Mathematics is a subject that requires logical thinking, problem-solving skills, and a deep understanding of various concepts. In this article, we present 100 math problems covering different topics, including algebra, geometry, calculus, and more. Each problem is followed by a step-by-step explanation of how to solve it, along with the correct answer. Let's dive in!

Problem 1:
Solve the equation: 3x + 5 = 17

Solution:
Step 1: Subtract 5 from both sides: 3x = 17 - 5
Step 2: Simplify: 3x = 12
Step 3: Divide both sides by 3: x = 12 ÷ 3
Step 4: Simplify: x = 4

Answer: x = 4

Problem 2:
Find the value of x in the equation: 2(3x - 1) = 10

Solution:
Step 1: Distribute the 2: 6x - 2 = 10
Step 2: Add 2 to both sides: 6x = 10 + 2
Step 3: Simplify: 6x = 12
Step 4: Divide both sides by 6: x = 12 ÷ 6
Step 5: Simplify: x = 2

Answer: x = 2

Problem 3:
Solve the following inequality: 2x + 3 > 9

Solution:
Step 1: Subtract 3 from both sides: 2x > 9 - 3
Step 2: Simplify: 2x > 6
Step 3: Divide both sides by 2 (since 2 is positive): x > 6 ÷ 2
Step 4: Simplify: x > 3

Answer: x > 3

Problem 4:
Evaluate the expression: 4 + 2 × 3 - 1

Solution:
Step 1: Perform multiplication: 4 + 6 - 1
Step 2: Perform addition and subtraction (left to right): 10 - 1
Step 3: Simplify: 9

Answer: 9

Problem 5:
Factorize the quadratic expression: x² - 4x - 12

Solution:
Step 1: Find two numbers that multiply to -12 and add up to -4. In this case, -6 and 2 satisfy these conditions.
Step 2: Rewrite the middle term: x² - 6x + 2x - 12
Step 3: Group the terms and factor by grouping: x(x - 6) + 2(x - 6)
Step 4: Factor out the common binomial: (x + 2)(x - 6)

Answer: (x + 2)(x - 6)

Problem 6:
Find the area of a rectangle with length 8 cm and width 5 cm.

Solution:
Step 1: Use the formula for the area of a rectangle: Area = length × width
Step 2: Substitute the given values: Area = 8 cm × 5 cm
Step 3: Multiply: Area = 40 cm²

Answer: 40 cm²

Problem 7:
Calculate the volume of a sphere with radius 3 cm. (Use π ≈ 3.14)

Solution:
Step 1: Use the formula for the volume of a sphere: Volume = (4/3)πr³
Step 2: Substitute the given values: Volume = (4/3) × 3.14 × (3 cm)³
Step 3: Simplify: Volume = (4/3) × 3.14 × 27 cm³
Step 4: Multiply: Volume ≈ 113.04 cm³ (rounded to two decimal places)

Answer: Volume ≈ 113.04 cm³

Problem 8:
Solve the system of equations:
2x + 3y = 10
4x - 5y = 8

Solution:
Step 1: Multiply the first equation by 2: 4x + 6y = 20
Step 2: Add the second equation: (4x + 6y) + (4x - 5y) = 20 + 8
Step 3: Simplify: 8x + y = 28
Step 4: Solve for y: y = 28 - 8x
Step 5: Substitute the value of y into the first equation: 2x + 3(28 - 8x) = 10
Step 6: Simplify: 2x + 84 - 24x = 10
Step 7: Combine like terms: -22x + 84 = 10
Step 8: Subtract 84 from both sides: -22x = 10 - 84
Step 9: Simplify: -22x = -74
Step 10: Divide both sides by -22: x = -74 ÷ -22
Step 11: Simplify: x = 3.36 (rounded to two decimal places)
Step 12: Substitute the value of x into the equation y = 28 - 8x: y = 28 - 8(3.36)
Step 13: Simplify: y = 28 - 26.88
Step 14: Simplify further: y ≈ 1.12 (rounded to two decimal places)

Answer: x ≈ 3.36 and y ≈ 1.12

Problem 9:
Find the derivative of the function f(x) = 3x² + 2x - 1.

Solution:
Step 1: Use the power rule for derivatives: f'(x) = 2(3x²) + 1(2x) + 0
Step 2: Simplify: f'(x) = 6x² + 2x

Answer: f'(x) = 6x² + 2x

Problem 10:
Simplify the expression: √(64) - √(16)

Solution:
Step 1: Evaluate the square roots: 8 - 4
Step 2: Simplify: 4

Answer: 4

Problem 11:
Solve the trigonometric equation: sin(x) = 0.5

Solution:
Step 1: Take the inverse sine of both sides: x = arcsin(0.5)
Step 2: Use a calculator or reference table to find the angle: x ≈ 30°

Answer: x ≈ 30°

Problem 12:
Calculate the factorial of 6 (written as 6!).

Solution:
Step 1: Multiply 6 by all positive integers less than it: 6! = 6 × 5 × 4 × 3 × 2 × 1
Step 2: Simplify: 6! = 720

Answer: 6! = 720

Problem 13:
Determine the perimeter of a triangle with side lengths 5 cm, 7 cm, and 9 cm.

Solution:
Step 1: Add the lengths of all three sides: Perimeter = 5 cm + 7 cm + 9 cm
Step 2: Simplify: Perimeter = 21 cm

Answer: Perimeter = 21 cm

Problem 14:
Solve the logarithmic equation: log₂(x) + log₂(2x + 6) = 3

Solution:
Step 1: Combine the logarithms using the product rule: log₂(x(2x + 6)) = 3
Step 2: Simplify the product inside the logarithm: log₂(2x² + 6x) = 3
Step 3: Rewrite the equation in exponential form: 2x² + 6x = 2³
Step 4: Simplify the exponent: 2x² + 6x = 8
Step 5: Rearrange the equation: 2x² + 6x - 8 = 0
Step 6: Factorize: (x - 1)(2x + 8) = 0
Step 7: Set each factor equal to zero and solve: x - 1 = 0 or 2x + 8 = 0
Step 8: Solve for x: x = 1 or x = -4

Answer: x = 1 or x = -4

Problem 15:
Evaluate the integral: ∫(4x² + 3x + 2) dx

Solution:
Step 1: Apply the power rule for integration: ∫(4x² + 3x + 2) dx = (4/3)x³ + (3/2)x² + 2x + C

Answer: ∫(4x² + 3x + 2) dx = (4/3)x³ + (3/2)x² + 2x + C

Problem 16:
Find the equation of a line passing through the points (2, 5) and (4, 9).

Solution:
Step 1: Calculate the slope using the formula: m = (y₂ - y₁) / (x₂ - x₁)
Step 2: Substitute the coordinates: m = (9 - 5) / (4 - 2) = 4 / 2 = 2
Step 3: Use the point-slope form of a line: y - y₁ = m(x - x₁)
Step 4: Substitute the slope and one point: y - 5 = 2(x - 2)
Step 5: Simplify: y - 5 = 2x - 4
Step 6: Rearrange the equation: y = 2x + 1

Answer: The equation of the line is y = 2x + 1

Problem 17:
Solve the quadratic equation: 2x² - 5x - 3 = 0

Solution:
Step 1: Factorize or use the quadratic formula to find the roots of the equation.
Since factorizing is not possible in this case, we'll use the quadratic formula.
Step 2: Apply the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)
Substituting the values from the given equation: x = (5 ± √(5² - 4(2)(-3))) / (2(2))
Step 3: Simplify: x = (5 ± √(25 + 24)) / 4
x = (5 ± √(49)) / 4
x = (5 ± 7) / 4

For the positive root:
x₁ = (5 + 7) / 4
x₁ = 12 / 4
x₁ = 3

For the negative root:
x₂ = (5 - 7) / 4
x₂ = -2 / 4
x₂ = -0.5

Answer: x = 3 or x = -0.5

Problem 18:
Simplify the complex number expression: (2 + 3i) + (4 - 2i)

Solution:
Step 1: Add the real parts: 2 + 4 = 6
Step 2: Add the imaginary parts: 3i - 2i = i

Answer: (2 + 3i) + (4 - 2i) = 6 + i

Problem 19:
Find the median of the following set of numbers: 5, 3, 9, 1, 7

Solution:
Step 1: Arrange the numbers in ascending order: 1, 3, 5, 7, 9
Step 2: Determine the middle value: The median is the middle number, which is 5.

Answer: The median is 5.

Problem 20:
Calculate the probability of rolling a 6 on a fair six-sided die.

Solution:
Step 1: Determine the number of favorable outcomes: There is only one favorable outcome, rolling a 6.
Step 2: Determine the total number of possible outcomes: There are six possible outcomes, one for each side of the die.
Step 3: Calculate the probability: Probability = favorable outcomes / total outcomes = 1/6

Answer: The probability of rolling a 6 on a fair six-sided die is 1/6.

Problem 21:
Simplify the expression: log₅(125)

Solution:
Step 1: Determine the exponent that gives 125 when raised to the base 5: 5³ = 125
Step 2: Simplify: log₅(125) = 3

Answer: log₅(125) = 3

Problem 22:
Solve the exponential equation: 2^(x + 1) = 16

Solution:
Step 1: Rewrite 16 as a power of 2: 16 = 2^4
Step 2: Set the exponents equal to each other: x + 1 = 4
Step 3: Solve for x: x = 4 - 1
x = 3

Answer: x = 3

Problem 23:
Find the area of a trapezoid with bases of length 6 cm and 10 cm, and a height of 8 cm.

Solution:
Step 1: Use the formula for the area of a trapezoid: Area = (1/2) × (a + b) × h
Step 2: Substitute the given values: Area = (1/2) × (6 cm + 10 cm) × 8 cm
Step 3: Simplify: Area = (1/2) × 16 cm × 8 cm
Step 4: Multiply: Area = 64 cm²

Answer: The area of the trapezoid is 64 cm².

Problem 24:
Solve the matrix equation: [2 1] [x] = [5]
[3 4] [y] [8]

Solution:
Step 1: Multiply the matrix and the column vector: 2x + y = 5 and 3x + 4y = 8
Step 2: Solve the system of equations using any appropriate method (substitution, elimination, etc.):

Multiply the first equation by 3: 6x + 3y = 15
Multiply the second equation by 2: 6x + 8y = 16
Subtract the first equation from the second equation: 6x + 8y - (6x + 3y) = 16 - 15
Simplify: 6x + 8y - 6x - 3y = 1
Simplify further: 5y = 1
Solve for y: y = 1/5
Substitute the value of y into the first equation: 2x + (1/5) = 5
Simplify: 2x + 1/5 = 5
Subtract 1/5 from both sides: 2x = 5 - 1/5
Simplify: 2x = 25/5 - 1/5
Simplify further: 2x = 24/5
Divide both sides by 2: x = (24/5) / 2
Simplify: x = 24/10
Simplify further: x = 12/5
Answer: x = 12/5 and y = 1/5

Problem 25:
Evaluate the limit: lim(x → 0) (3x² + 2x + 1) / x

Solution:
Step 1: Substitute the value of x into the expression: (3(0)² + 2(0) + 1) / 0
Step 2: Simplify: (0 + 0 + 1) / 0
Step 3: Since the denominator is zero, the limit is undefined.

Answer: The limit does not exist.

Problem 26:
Simplify the expression: 4x - 2(3x + 5)

Solution:
Step 1: Distribute the -2: 4x - 6x - 10
Step 2: Combine like terms: -2x - 10

Answer: -2x - 10

Problem 27:
Find the value of x in the equation: 2(x - 3) + 5 = 17

Solution:
Step 1: Distribute the 2: 2x - 6 + 5 = 17
Step 2: Combine like terms: 2x - 1 = 17
Step 3: Add 1 to both sides: 2x = 18
Step 4: Divide both sides by 2: x = 9

Answer: x = 9

Problem 28:
Solve the inequality: 3x + 7 > 4x - 5

Solution:
Step 1: Subtract 3x from both sides: 7 > x - 5
Step 2: Add 5 to both sides: 12 > x

Answer: x < 12

Problem 29:
Evaluate the expression: 3(2² - 1) + 4

Solution:
Step 1: Simplify the exponent: 3(4 - 1) + 4
Step 2: Simplify the parentheses: 3(3) + 4
Step 3: Multiply: 9 + 4

Answer: 13

Problem 30:
Factorize the quadratic expression: x² + 7x + 10

Solution:
Step 1: Find two numbers that multiply to 10 and add up to 7. In this case, 2 and 5 satisfy these conditions.
Step 2: Rewrite the middle term: x² + 2x + 5x + 10
Step 3: Group the terms and factor by grouping: (x² + 2x) + (5x + 10)
Step 4: Factor out the common binomial: x(x + 2) + 5(x + 2)
Step 5: Combine like terms: (x + 2)(x + 5)

Answer: (x + 2)(x + 5)

Problem 31:
Calculate the perimeter of a square with side length 9 cm.

Solution:
Step 1: Use the formula for the perimeter of a square: Perimeter = 4 × side length
Step 2: Substitute the given value: Perimeter = 4 × 9 cm

Answer: Perimeter = 36 cm

Problem 32:
Determine the volume of a cylinder with radius 5 cm and height 10 cm. (Use π ≈ 3.14)

Solution:
Step 1: Use the formula for the volume of a cylinder: Volume = πr²h
Step 2: Substitute the given values: Volume = 3.14 × (5 cm)² × 10 cm
Step 3: Simplify: Volume = 3.14 × 25 cm² × 10 cm
Step 4: Multiply: Volume = 785 cm³

Answer: Volume = 785 cm³

Problem 33:
Solve the system of equations:
2x + 3y = 10
5x - 2y = 7

Solution:
Step 1: Multiply the first equation by 2: 4x + 6y = 20
Step 2: Multiply the second equation by 3: 15x - 6y = 21
Step 3: Add the equations: (4x + 6y) + (15x - 6y) = 20 + 21
Step 4: Simplify: 19x = 41
Step 5: Divide both sides by 19: x = 41/19
Step 6: Substitute the value of x into the first equation: 2(41/19) + 3y = 10
Step 7: Simplify: 82/19 + 3y = 10
Step 8: Subtract 82/19 from both sides: 3y = 10 - 82/19
Step 9: Simplify: 3y = 190/19 - 82/19
Step 10: Combine like terms: 3y = 108/19
Step 11: Divide both sides by 3: y = 108/57
Step 12: Simplify: y = 6/19

Answer: x = 41/19 and y = 6/19

Problem 34:
Find the derivative of the function f(x) = 4x³ + 2x² - 1.

Solution:
Step 1: Take the derivative of each term using the power rule:
f'(x) = 4(3x²) + 2(2x) - 0
Step 2: Simplify: f'(x) = 12x² + 4x

Answer: f'(x) = 12x² + 4x

Problem 35:
Simplify the expression: √(81) - √(25)

Solution:
Step 1: Evaluate the square roots: 9 - 5
Step 2: Simplify: 4

Answer: 4

Problem 36:
Solve the trigonometric equation: cos(x) = 0.5

Solution:
Step 1: Take the inverse cosine of both sides: x = arccos(0.5)
Step 2: Use a calculator or reference table to find the angle: x = π/3 or x = 2π/3

Answer: x = π/3 or x = 2π/3

Problem 37:
Calculate the probability of rolling a 6 on a fair six-sided die.

Solution:
Step 1: Determine the number of favorable outcomes: There is only one favorable outcome, rolling a 6.
Step 2: Determine the total number of possible outcomes: There are six possible outcomes, one for each side of the die.
Step 3: Calculate the probability: Probability = favorable outcomes / total outcomes = 1/6

Answer: The probability of rolling a 6 on a fair six-sided die is 1/6.

Problem 38:
Simplify the expression: log₅(125)

Solution:
Step 1: Determine the exponent that gives 125 when raised to the base 5: 5³ = 125
Step 2: Simplify: log₅(125) = 3

Answer: log₅(125) = 3

Problem 39:
Solve the exponential equation: 2^(x + 1) = 16

Solution:
Step 1: Rewrite 16 as a power of 2: 16 = 2^4
Step 2: Set the exponents equal to each other: x + 1 = 4
Step 3: Solve for x: x = 4 - 1
x = 3

Answer: x = 3

Problem 40:
Find the area of a trapezoid with bases of length 6 cm and 10 cm, and a height of 8 cm.

Solution:
Step 1: Use the formula for the area of a trapezoid: Area = (1/2) × (a + b) × h
Step 2: Substitute the given values: Area = (1/2) × (6 cm + 10 cm) × 8 cm
Step 3: Simplify: Area = (1/2) × 16 cm × 8 cm
Step 4: Multiply: Area = 64 cm²

Answer: The area of the trapezoid is 64 cm².

Problem 41:
Solve the matrix equation: [2 1] [x] = [5]
[3 4] [y] [8]

Solution:
Step 1: Multiply the inverse of the matrix on both sides to isolate the variables.

[2 1]⁻¹ [2 1] [x] = [2 1]⁻¹ [5]
[3 4] [y] [3 4] [8]

[x] = [2 1]⁻¹ [5]
[y] [3 4] [8]

Step 2: Calculate the inverse of the matrix [2 1]
[3 4]

The inverse of the matrix is:

[ 4 -1]
[-3 2]

Step 3: Multiply the inverse of the matrix by the right-hand side:

[x] = [ 4 -1] [5]
[y] [-3 2] [8]

Step 4: Simplify:

[x] = [4(5) + (-1)(8)]
[y] [-3(5) + 2(8)]

[x] = [20 - 8]
[y] [-15 + 16]

[x] = [12]
[y] [ 1]

Answer: x = 12 and y = 1

Problem 42:
Evaluate the limit: lim(x → 0) (3x² + 2x + 1) / x

Solution:
Step 1: Substitute the value of x into the expression: (3(0)² + 2(0) + 1) / 0
Step 2: Simplify: (0 + 0 + 1) / 0
Step 3: Since the denominator is zero, the limit is undefined.

Answer: The limit does not exist.

Problem 43:
Solve the logarithmic equation: log₂(x) + log₂(2x + 6) = 3

Solution:
Step 1: Combine the logarithms using the product rule: log₂(x(2x + 6)) = 3
Step 2: Simplify the product inside the logarithm: log₂(2x² + 6x) = 3
Step 3: Rewrite the equation in exponential form: 2x² + 6x = 2³
Step 4: Simplify the exponent: 2x² + 6x = 8
Step 5: Rearrange the equation: 2x² + 6x - 8 = 0
Step 6: Factorize: (x - 1)(2x + 4) = 0
Step 7: Set each factor equal to zero and solve: x - 1 = 0 or 2x + 4 = 0
Step 8: Solve for x: x = 1 or x = -2

Answer: x = 1 or x = -2

Problem 44:
Find the area of a circle with a radius of 6 cm. (Use π ≈ 3.14)

Solution:
Step 1: Use the formula for the area of a circle: Area = πr²
Step 2: Substitute the given value: Area = 3.14 × (6 cm)²
Step 3: Simplify: Area = 3.14 × 36 cm²
Step 4: Multiply: Area = 113.04 cm²

Answer: The area of the circle is 113.04 cm².

Problem 45:
Solve the equation: 2sin(x) = 1

Solution:
Step 1: Divide both sides by 2: sin(x) = 1/2
Step 2: Use a calculator or reference table to find the angles where sin(x) = 1/2: x = π/6 or x = 5π/6

Answer: x = π/6 or x = 5π/6


Problem 46:
Simplify the expression: (3x + 2y) - (2x - y)

Solution:
Step 1: Distribute the negative sign: 3x + 2y - 2x + y
Step 2: Combine like terms: (3x - 2x) + (2y + y)
Step 3: Simplify: x + 3y

Answer: x + 3y

Problem 47:
Find the value of x in the equation: 5(x - 2) = 3(2x + 1)

Solution:
Step 1: Distribute the 5 and 3: 5x - 10 = 6x + 3
Step 2: Subtract 5x and 6x from both sides: -10 - 3 = 6x - 5x
Step 3: Simplify: -13 = x

Answer: x = -13

Problem 48:
Solve the inequality: 2x + 3 < 5x - 2

Solution:
Step 1: Subtract 2x from both sides: 3 < 3x - 2
Step 2: Add 2 to both sides: 5 < 3x
Step 3: Divide both sides by 3: 5/3 < x

Answer: x > 5/3

Problem 49:
Evaluate the expression: 2² + 3³ - 4⁴

Solution:
Step 1: Evaluate the exponents: 2² + 3³ - 4⁴ = 4 + 27 - 256
Step 2: Simplify: 4 + 27 - 256 = -225

Answer: -225

Problem 50:
Factorize the quadratic expression: 2x² + 5x + 3

Solution:
Step 1: Find two numbers that multiply to 6 and add up to 5. In this case, 2 and 3 satisfy these conditions.
Step 2: Rewrite the middle term: 2x² + 2x + 3x + 3
Step 3: Group the terms and factor by grouping: (2x² + 2x) + (3x + 3)
Step 4: Factor out the common binomial: 2x(x + 1) + 3(x + 1)
Step 5: Combine like terms: (2x + 3)(x + 1)

Answer: (2x + 3)(x + 1)

Problem 51:
Calculate the perimeter of a rectangle with length 12 cm and width 5 cm.

Solution:
Step 1: Use the formula for the perimeter of a rectangle: Perimeter = 2(length + width)
Step 2: Substitute the given values: Perimeter = 2(12 cm + 5 cm)
Step 3: Simplify: Perimeter = 2(17 cm)
Step 4: Multiply: Perimeter = 34 cm

Answer: The perimeter of the rectangle is 34 cm.

Problem 52:
Determine the volume of a cone with radius 4 cm and height 8 cm. (Use π ≈ 3.14)

Solution:
Step 1: Use the formula for the volume of a cone: Volume = (1/3) × πr²h
Step 2: Substitute the given values: Volume = (1/3) × 3.14 × (4 cm)² × 8 cm
Step 3: Simplify: Volume = (1/3) × 3.14 × 16 cm² × 8 cm
Step 4: Multiply: Volume = (1/3) × 3.14 × 128 cm³
Step 5: Simplify further: Volume ≈ 134.19 cm³ (rounded to two decimal places)

Answer: The volume of the cone is approximately 134.19 cm³.

Problem 53:
Solve the system of equations:
3x + 2y = 10
4x - 5y = -3

Solution:
Step 1: Multiply the first equation by 4: 12x + 8y = 40
Step 2: Multiply the second equation by 3: 12x - 15y = -9
Step 3: Subtract the second equation from the first equation: (12x + 8y) - (12x - 15y) = 40 - (-9)
Step 4: Simplify: 12x + 8y - 12x + 15y = 40 + 9
Step 5: Combine like terms: 23y = 49
Step 6: Divide both sides by 23: y = 49/23
Step 7: Substitute the value of y into the first equation: 3x + 2(49/23) = 10
Step 8: Simplify: 3x + 98/23 = 10
Step 9: Subtract 98/23 from both sides: 3x = 10 - 98/23
Step 10: Simplify: 3x = 230/23 - 98/23
Step 11: Combine like terms: 3x = 132/23
Step 12: Divide both sides by 3: x = (132/23) / 3
Step 13: Simplify: x = 132/69
Step 14: Simplify further: x = 44/23

Answer: x = 44/23 and y = 49/23

Problem 54:
Find the derivative of the function f(x) = sin(x) + cos(x).

Solution:
Step 1: Take the derivative of each term using the sum rule:
f'(x) = cos(x) - sin(x)

Answer: f'(x) = cos(x) - sin(x)

Problem 55:
Simplify the expression: √(36) + √(49)

Solution:
Step 1: Evaluate the square roots: 6 + 7
Step 2: Simplify: 13

Answer: 13

Problem 56:
Solve the trigonometric equation: tan(x) = 1

Solution:
Step 1: Take the inverse tangent of both sides: x = arctan(1)
Step 2: Use a calculator or reference table to find the angle: x = π/4

Answer: x = π/4

Problem 57:
Calculate the factorial of 7 (written as 7!).

Solution:
Step 1: Multiply 7 by all positive integers less than it: 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1
Step 2: Simplify: 7! = 5040

Answer: 7! = 5040

Problem 58:
Determine the area of a parallelogram with base length 6 cm and height 8 cm.

Solution:
Step 1: Use the formula for the area of a parallelogram: Area = base × height
Step 2: Substitute the given values: Area = 6 cm × 8 cm
Step 3: Multiply: Area = 48 cm²

Answer: The area of the parallelogram is 48 cm².

Problem 59:
Solve the logarithmic equation: log₃(x) + log₃(x - 4) = 2

Solution:
Step 1: Combine the logarithms using the product rule: log₃(x(x - 4)) = 2
Step 2: Simplify the product inside the logarithm: log₃(x² - 4x) = 2
Step 3: Rewrite the equation in exponential form: 3² = x² - 4x
Step 4: Simplify the exponent: 9 = x² - 4x
Step 5: Rearrange the equation: x² - 4x - 9 = 0
Step 6: Solve the quadratic equation using factoring, completing the square, or the quadratic formula:

Factoring: (x - 3)(x + 3) = 0
Setting each factor equal to zero: x - 3 = 0 or x + 3 = 0
Solving for x: x = 3 or x = -3
Answer: x = 3 or x = -3

Problem 60:
Evaluate the integral: ∫(2x + 5) dx

Solution:
Step 1: Apply the power rule for integration: ∫(2x + 5) dx = x² + 5x + C

Answer: ∫(2x + 5) dx = x² + 5x + C (where C is the constant of integration)

Problem 61:
Find the equation of the line passing through the points (-1, 3) and (2, -4).

Solution:
Step 1: Calculate the slope using the formula: m = (y₂ - y₁) / (x₂ - x₁)
Step 2: Substitute the coordinates: m = (-4 - 3) / (2 - (-1)) = -7 / 3
Step 3: Use the point-slope form of a line: y - y₁ = m(x - x₁)
Step 4: Substitute the slope and one point: y - 3 = (-7/3)(x - (-1))
Step 5: Simplify: y - 3 = (-7/3)(x + 1)
Step 6: Rearrange the equation: y = (-7/3)x - 7/3 + 9/3
Step 7: Simplify: y = (-7/3)x + 2/3

Answer: The equation of the line is y = (-7/3)x + 2/3

Problem 62:
Solve the quadratic equation: x² - 6x + 5 = 0

Solution:
Step 1: Factorize or use the quadratic formula to find the roots of the equation.

Factoring: (x - 5)(x - 1) = 0
Setting each factor equal to zero: x - 5 = 0 or x - 1 = 0
Solving for x: x = 5 or x = 1
Answer: x = 5 or x = 1

​Problem 63:
Simplify the expression: 2(x - 3) - 3(2x + 1)

Solution:
Step 1: Distribute the 2 and -3: 2x - 6 - 6x - 3
Step 2: Combine like terms: (2x - 6x) + (-6 - 3)
Step 3: Simplify: -4x - 9

Answer: -4x - 9

Problem 64:
Find the value of x in the equation: 4(x + 2) = 3(x - 1) + 5

Solution:
Step 1: Distribute the 4 and 3: 4x + 8 = 3x - 3 + 5
Step 2: Combine like terms: 4x + 8 = 3x + 2
Step 3: Subtract 3x from both sides: x + 8 = 2
Step 4: Subtract 8 from both sides: x = 2 - 8
Step 5: Simplify: x = -6

Answer: x = -6

Problem 65:
Solve the inequality: 3x + 5 ≥ 2x - 3

Solution:
Step 1: Subtract 2x from both sides: x + 5 ≥ -3
Step 2: Subtract 5 from both sides: x ≥ -8

Answer: x ≥ -8

Problem 66:
Evaluate the expression: 5² - 4(3 + 1)

Solution:
Step 1: Simplify within parentheses: 5² - 4(4)
Step 2: Evaluate the exponent: 5² = 25
Step 3: Multiply: 25 - 4(4)
Step 4: Simplify further: 25 - 16
Step 5: Subtract: 9

Answer: 9

Problem 67:
Factorize the quadratic expression: 3x² + 7x - 2

Solution:
Step 1: Find two numbers that multiply to -6 and add up to 7. In this case, 8 and -1 satisfy these conditions.
Step 2: Rewrite the middle term: 3x² + 8x - x - 2
Step 3: Group the terms and factor by grouping: (3x² + 8x) + (-x - 2)
Step 4: Factor out the common binomial: x(3x + 8) - 1(3x + 8)
Step 5: Combine like terms: (x - 1)(3x + 8)

Answer: (x - 1)(3x + 8)

Problem 68:
Calculate the perimeter of a triangle with side lengths 4 cm, 6 cm, and 7 cm.

Solution:
Step 1: Add the lengths of all three sides: 4 cm + 6 cm + 7 cm
Step 2: Add: 17 cm

Answer: The perimeter of the triangle is 17 cm.

Problem 69:
Determine the volume of a rectangular prism with length 8 cm, width 5 cm, and height 3 cm.

Solution:
Step 1: Use the formula for the volume of a rectangular prism: Volume = length × width × height
Step 2: Substitute the given values: Volume = 8 cm × 5 cm × 3 cm
Step 3: Multiply: Volume = 120 cm³

Answer: The volume of the rectangular prism is 120 cm³.

Problem 70:
Solve the system of equations:
2x + y = 5
3x - 2y = 4

Solution:
Step 1: Multiply the first equation by 2: 4x + 2y = 10
Step 2: Add the equations: (4x + 2y) + (3x - 2y) = 10 + 4
Step 3: Simplify: 7x = 14
Step 4: Divide both sides by 7: x = 2
Step 5: Substitute the value of x into the first equation: 2(2) + y = 5
Step 6: Simplify: 4 + y = 5
Step 7: Subtract 4 from both sides: y = 5 - 4
Step 8: Simplify: y = 1

Answer: x = 2 and y = 1

Problem 71:
Find the derivative of the function f(x) = e^x + 3x².

Solution:
Step 1: Take the derivative of each term using the sum rule and the power rule:
f'(x) = e^x + 6x

Answer: f'(x) = e^x + 6x

Problem 72:
Simplify the expression: log₄(16) + log₄(2)

Solution:
Step 1: Evaluate the logarithms: 2 + 1
Step 2: Simplify: 3

Answer: 3

Problem 73:
Solve the trigonometric equation: sin(x) = 0.5

Solution:
Step 1: Take the inverse sine of both sides: x = arcsin(0.5)
Step 2: Use a calculator or reference table to find the angle: x = π/6 or x = 5π/6

Answer: x = π/6 or x = 5π/6

Problem 74:
Calculate the factorial of 6 (written as 6!).

Solution:
Step 1: Multiply 6 by all positive integers less than it: 6! = 6 × 5 × 4 × 3 × 2 × 1
Step 2: Simplify: 6! = 720

Answer: 6! = 720

Problem 75:
Determine the area of a trapezoid with bases of length 10 cm and 6 cm, and a height of 8 cm.

Solution:
Step 1: Use the formula for the area of a trapezoid: Area = (1/2) × (a + b) × h
Step 2: Substitute the given values: Area = (1/2) × (10 cm + 6 cm) × 8 cm
Step 3: Simplify: Area = (1/2) × 16 cm × 8 cm
Step 4: Multiply: Area = 64 cm²

Answer: The area of the trapezoid is 64 cm².

Problem 76:
Simplify the expression: cos²(x) - sin²(x)

Solution:
Step 1: Use the trigonometric identity cos²(x) - sin²(x) = cos(2x).
Step 2: Simplify the expression: cos²(x) - sin²(x) = cos(2x).

Answer: cos²(x) - sin²(x) simplifies to cos(2x).

Problem 77:
Find the equation of the line perpendicular to the line 2x - 3y = 5 and passing through the point (4, 2).

Solution:
Step 1: Find the slope of the given line by rearranging it in slope-intercept form: 2x - 3y = 5 -> -3y = -2x + 5 -> y = (2/3)x - 5/3
Step 2: Determine the slope of the perpendicular line by taking the negative reciprocal of (2/3): m = -3/2
Step 3: Use the point-slope form of a line with the given point: y - 2 = (-3/2)(x - 4)
Step 4: Simplify: y - 2 = (-3/2)x + 6
Step 5: Rearrange the equation: y = (-3/2)x + 8

Answer: The equation of the line perpendicular to 2x - 3y = 5 and passing through the point (4, 2) is y = (-3/2)x + 8.

Problem 78:
Solve the quadratic equation: 2x² + 5x - 3 = 0

Solution:
Step 1: Factorize or use the quadratic formula to find the roots of the equation.

Factoring: (2x - 1)(x + 3) = 0
Setting each factor equal to zero: 2x - 1 = 0 or x + 3 = 0
Solving for x: x = 1/2 or x = -3
Answer: x = 1/2 or x = -3

Problem 79:
Evaluate the integral: ∫(3x² - 2x + 4) dx

Solution:
Step 1: Apply the power rule for integration: ∫(3x² - 2x + 4) dx = x³ - x² + 4x + C

Answer: ∫(3x² - 2x + 4) dx = x³ - x² + 4x + C (where C is the constant of integration)

Problem 80:
Find the equation of the circle with center (-2, 3) and radius 5.

Solution:
Step 1: Use the standard form of the equation for a circle: (x - h)² + (y - k)² = r²
Step 2: Substitute the given values: (x - (-2))² + (y - 3)² = 5²
Step 3: Simplify: (x + 2)² + (y - 3)² = 25

Answer: The equation of the circle is (x + 2)² + (y - 3)² = 25.

Problem 81:
Simplify the expression: 3! + 4! - 2!

Solution:
Step 1: Evaluate the factorials: 3! = 3 × 2 × 1 = 6, 4! = 4 × 3 × 2 × 1 = 24, 2! = 2 × 1 = 2
Step 2: Simplify: 6 + 24 - 2 = 28

Answer: 3! + 4! - 2! simplifies to 28.

Problem 82:
Find the value of x in the equation: log₂(x) = 5

Solution:
Step 1: Rewrite the equation in exponential form: 2⁵ = x
Step 2: Evaluate the exponent: 32 = x

Answer: x = 32

Problem 83:
Solve the trigonometric equation: cos²(x) + sin²(x) = 1

Solution:
Step 1: Use the Pythagorean identity for trigonometric functions: cos²(x) + sin²(x) = 1

Answer: The equation cos²(x) + sin²(x) = 1 holds true for all values of x.

Problem 84:
Calculate the perimeter of a square with side length 10 cm.

Solution:
Step 1: Use the formula for the perimeter of a square: Perimeter = 4 × side length
Step 2: Substitute the given value: Perimeter = 4 × 10 cm
Step 3: Multiply: Perimeter = 40 cm

Answer: The perimeter of the square is 40 cm.

Problem 85:
Determine the volume of a sphere with radius 3 cm. (Use π ≈ 3.14)

Solution:
Step 1: Use the formula for the volume of a sphere: Volume = (4/3) × π × radius³
Step 2: Substitute the given value: Volume = (4/3) × 3.14 × (3 cm)³
Step 3: Simplify: Volume = (4/3) × 3.14 × 27 cm³
Step 4: Multiply: Volume ≈ 113.04 cm³ (rounded to two decimal places)

Answer: The volume of the sphere is approximately 113.04 cm³.

Problem 86:
Solve the system of equations:
2x + 3y = 7
4x - 5y = 1

Solution:
Step 1: Multiply the first equation by 2: 4x + 6y = 14
Step 2: Multiply the second equation by 3: 12x - 15y = 3
Step 3: Add the equations: (4x + 6y) + (12x - 15y) = 14 + 3
Step 4: Simplify: 16x - 9y = 17
Step 5: Solve for y in terms of x: -9y = 17 - 16x
Step 6: Divide both sides by -9: y = (16x - 17)/9
Step 7: Substitute the value of y into the first equation: 2x + 3[(16x - 17)/9] = 7
Step 8: Simplify: 2x + (48x - 51)/9 = 7
Step 9: Multiply through by 9 to eliminate the denominator: 18x + 48x - 51 = 63
Step 10: Combine like terms: 66x - 51 = 63
Step 11: Add 51 to both sides: 66x = 114
Step 12: Divide both sides by 66: x = 114/66
Step 13: Simplify: x = 19/11

Answer: x = 19/11, y = (16(19/11) - 17)/9

Problem 87:
Find the derivative of the function f(x) = 5x³ - 2x² + 3x - 1.

Solution:
Step 1: Take the derivative of each term using the power rule:
f'(x) = (3)(5x²) - (2)(2x) + 3
Step 2: Simplify: f'(x) = 15x² - 4x + 3

Answer: f'(x) = 15x² - 4x + 3

Problem 88:
Simplify the expression: √(64) - √(9)

Solution:
Step 1: Evaluate the square roots: 8 - 3
Step 2: Simplify: 5

Answer: 5

Problem 89:
Solve the equation: 2x + 5 = 3x - 1

Solution:
Step 1: Subtract 2x from both sides: 5 = x - 1
Step 2: Add 1 to both sides: 6 = x

Answer: x = 6

Problem 90:
Find the equation of the line parallel to the line 3x + 2y = 10 and passing through the point (1, 4).

Solution:
Step 1: Rewrite the given equation in slope-intercept form: 2y = -3x + 10 -> y = (-3/2)x + 5
Step 2: Determine the slope of the given line: The coefficient of x is -3/2, so the slope is -3/2.
Step 3: The line parallel to this line will have the same slope, -3/2.
Step 4: Use the point-slope form of a line with the given point: y - 4 = (-3/2)(x - 1)
Step 5: Simplify: y - 4 = (-3/2)x + 3/2
Step 6: Rearrange the equation: y = (-3/2)x + 3/2 + 4
Step 7: Simplify further: y = (-3/2)x + 11/2

Answer: The equation of the line parallel to 3x + 2y = 10 and passing through the point (1, 4) is y = (-3/2)x + 11/2.

Problem 91:
Solve the quadratic equation: x² - 8x + 12 = 0

Solution:
Step 1: Factorize or use the quadratic formula to find the roots of the equation.

Factoring: (x - 2)(x - 6) = 0
Setting each factor equal to zero: x - 2 = 0 or x - 6 = 0
Solving for x: x = 2 or x = 6
Answer: x = 2 or x = 6

Problem 92:
Evaluate the integral: ∫(2cos(x) - 3sin(x)) dx

Solution:
Step 1: Apply the integral rules for cosine and sine functions: ∫(2cos(x) - 3sin(x)) dx = 2∫cos(x) dx - 3∫sin(x) dx
Step 2: Evaluate the integrals using the antiderivatives: 2sin(x) + 3cos(x) + C

Answer: ∫(2cos(x) - 3sin(x)) dx = 2sin(x) + 3cos(x) + C (where C is the constant of integration)

Problem 93:
Find the equation of the parabola with vertex (-2, 3) and focus (-2, 5).

Solution:
Step 1: The parabola is vertical, so the equation takes the form (x - h)² = 4p(y - k), where (h, k) is the vertex.
Step 2: Substitute the given values: (x + 2)² = 4p(y - 3)
Step 3: Since the focus is two units above the vertex, the distance between the focus and vertex is 2p = 2.
Step 4: Solve for p: p = 1
Step 5: Substitute the value of p into the equation: (x + 2)² = 4(y - 3)
Step 6: Simplify: (x + 2)² = 4y - 12
Step 7: Expand and rearrange the equation: x² + 4x + 4 = 4y
Step 8: Divide both sides by 4: (1/4)x² + x + 1 = y

Answer: The equation of the parabola is y = (1/4)x² + x + 1.

Problem 94:
Simplify the expression: 3³ - 2² + 5¹

Solution:
Step 1: Evaluate the exponents and addition: 27 - 4 + 5
Step 2: Simplify: 28

Answer: 28

Problem 95:
Find the value of x in the equation: 4(x - 1) = 3(x + 2)

Solution:
Step 1: Distribute the 4 and 3: 4x - 4 = 3x + 6
Step 2: Subtract 3x from both sides: x - 4 = 6
Step 3: Add 4 to both sides: x = 10

Answer: x = 10

Problem 96:
Solve the inequality: 2x + 5 > 3x - 1

Solution:
Step 1: Subtract 2x from both sides: 5 > x - 1
Step 2: Add 1 to both sides: 6 > x

Answer: x < 6

Problem 97:
Evaluate the expression: 2³ + 3² - 4¹

Solution:
Step 1: Evaluate the exponents and addition: 8 + 9 - 4
Step 2: Simplify: 13

Answer: 13

Problem 98:
Factorize the quadratic expression: x² - 9

Solution:
Step 1: Recognize the difference of squares pattern: x² - 9 = (x - 3)(x + 3)

Answer: (x - 3)(x + 3)

Problem 99:
Calculate the area of a triangle with base length 5 cm and height 8 cm.

Solution:
Step 1: Use the formula for the area of a triangle: Area = (1/2) × base × height
Step 2: Substitute the given values: Area = (1/2) × 5 cm × 8 cm
Step 3: Multiply: Area = 20 cm²

Answer: The area of the triangle is 20 cm².

Problem 100:
Solve the logarithmic equation: log₄(x - 1) + log₄(x + 1) = 2

Solution:
Step 1: Combine the logarithms using the product rule: log₄((x - 1)(x + 1)) = 2
Step 2: Simplify the product inside the logarithm: log₄(x² - 1) = 2
Step 3: Rewrite the equation in exponential form: 4² = x² - 1
Step 4: Evaluate the exponent: 16 = x² - 1
Step 5: Add 1 to both sides: 17 = x²
Step 6: Take the square root of both sides: x = ±√17

Answer: x = ±√17