Math Problems & Solutions — Worked Examples

A library of common math problems with step-by-step solutions, taught the way Mr. Neal teaches them in tutoring sessions. Use these to study, check your work, or learn a method. If you get stuck, book a free consultation and we’ll work through it live.

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Algebra Problems

Linear equations, quadratics, systems, and word problems — the core algebra most students need help with.

Problem 1 — Solving a Linear Equation

Solve: 3x + 7 = 22

Step 1. Subtract 7 from both sides to isolate the variable term:
3x + 7 − 7 = 22 − 7
3x = 15

Step 2. Divide both sides by 3:
x = 15 ÷ 3
x = 5
Answer: x = 5

Problem 2 — Solving a Quadratic by Factoring

Solve: x² + 5x + 6 = 0

Step 1. Look for two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3.

Step 2. Factor the quadratic:
(x + 2)(x + 3) = 0

Step 3. Set each factor equal to zero (Zero Product Property):
x + 2 = 0 → x = −2
x + 3 = 0 → x = −3
Answer: x = −2 or x = −3

Problem 3 — Quadratic Formula

Solve: 2x² + 3x − 5 = 0

Step 1. Identify a, b, c: a = 2, b = 3, c = −5.

Step 2. Apply the quadratic formula x = [−b ± √(b² − 4ac)] / (2a):
x = [−3 ± √(9 − 4·2·(−5))] / (2·2)
x = [−3 ± √(9 + 40)] / 4
x = [−3 ± √49] / 4
x = [−3 ± 7] / 4

Step 3. Two solutions:
x = (−3 + 7) / 4 = 1
x = (−3 − 7) / 4 = −5/2
Answer: x = 1 or x = −5/2

Problem 4 — System of Linear Equations (Substitution)

Solve: y = 2x + 1 and 3x + y = 11

Step 1. Substitute y = 2x + 1 into the second equation:
3x + (2x + 1) = 11
5x + 1 = 11
5x = 10
x = 2

Step 2. Plug x = 2 back into y = 2x + 1:
y = 2(2) + 1 = 5
Answer: (x, y) = (2, 5)

Problem 5 — Word Problem (Rate and Distance)

Question: A car travels at 60 mph for 2.5 hours. How far does it travel?

Step 1. Use the distance formula: distance = rate × time.

Step 2. Substitute the values:
distance = 60 mph × 2.5 hr = 150 miles
Answer: 150 miles
Tutoring tip: When you see a word problem, write down what you know (the “givens”) and what you’re looking for before doing any algebra. That clarity is half the battle.

Geometry Problems

Area, perimeter, the Pythagorean theorem, and angle relationships — the geometry that shows up in homework and on tests.

Problem 6 — Pythagorean Theorem

Question: A right triangle has legs of 6 and 8. What is the length of the hypotenuse?

Step 1. Apply a² + b² = c²:
6² + 8² = c²
36 + 64 = c²
100 = c²

Step 2. Take the square root of both sides:
c = 10
Answer: hypotenuse = 10

Problem 7 — Area of a Triangle

Question: Find the area of a triangle with base 12 and height 5.

Step 1. Use the triangle area formula: A = (1/2) × base × height.

Step 2. Substitute:
A = (1/2) × 12 × 5 = 30
Answer: Area = 30 square units

Problem 8 — Circumference and Area of a Circle

Question: A circle has radius 7. Find the circumference and the area. (Use π ≈ 3.14.)

Step 1. Circumference C = 2πr = 2 × 3.14 × 7 = 43.96
Step 2. Area A = πr² = 3.14 × 49 = 153.86
Answer: C ≈ 43.96, A ≈ 153.86

Problem 9 — Sum of Interior Angles of a Polygon

Question: What is the sum of interior angles of a hexagon (6-sided polygon)?

Step 1. Use the formula (n − 2) × 180°, where n is the number of sides.

Step 2. Substitute n = 6:
(6 − 2) × 180° = 4 × 180° = 720°
Answer: 720°

Pre-Calculus Problems

Functions, logarithms, and trig identities — the bridge between algebra and calculus.

Problem 10 — Evaluating a Function

Given: f(x) = x² − 3x + 2. Find f(4).

Step 1. Substitute x = 4 into the function:
f(4) = 4² − 3(4) + 2
f(4) = 16 − 12 + 2
f(4) = 6
Answer: f(4) = 6

Problem 11 — Solving a Logarithmic Equation

Solve: log₂(x) = 5

Step 1. Convert the log equation to its exponential form:
log₂(x) = 5 means 2⁵ = x

Step 2. Evaluate 2⁵:
x = 32
Answer: x = 32

Problem 12 — Pythagorean Identity

Given: sin(θ) = 3/5 and θ is in the first quadrant. Find cos(θ).

Step 1. Use the Pythagorean identity sin²(θ) + cos²(θ) = 1:
(3/5)² + cos²(θ) = 1
9/25 + cos²(θ) = 1
cos²(θ) = 16/25

Step 2. Take the positive square root (first quadrant, cosine is positive):
cos(θ) = 4/5
Answer: cos(θ) = 4/5

Calculus Problems

Limits, derivatives, integrals — the core of AP Calculus and Calc I.

Problem 13 — Finding a Derivative (Power Rule)

Find dy/dx if: y = 3x⁴ − 2x² + 7x − 5

Step 1. Apply the power rule term by term: d/dx(xⁿ) = n·xⁿ⁻¹.

Step 2. Differentiate each term:
d/dx(3x⁴) = 12x³
d/dx(−2x²) = −4x
d/dx(7x) = 7
d/dx(−5) = 0
Answer: dy/dx = 12x³ − 4x + 7

Problem 14 — Limit at a Point

Evaluate: lim (x → 3) (x² − 9) / (x − 3)

Step 1. Direct substitution gives 0/0, so factor the numerator:
x² − 9 = (x + 3)(x − 3)

Step 2. Cancel the common factor (x − 3):
(x + 3)(x − 3) / (x − 3) = x + 3

Step 3. Substitute x = 3:
3 + 3 = 6
Answer: Limit = 6

Problem 15 — Definite Integral

Evaluate: ∫ from 0 to 2 of (3x² + 2x) dx

Step 1. Find the antiderivative:
∫ 3x² dx = x³
∫ 2x dx = x²
So the antiderivative is x³ + x².

Step 2. Evaluate at the bounds (Fundamental Theorem of Calculus):
[x³ + x²] from 0 to 2 = (8 + 4) − (0 + 0) = 12
Answer: ∫ = 12

Problem 16 — Product Rule

Find the derivative of: y = x² · sin(x)

Step 1. Apply the product rule: (fg)’ = f’g + fg’.
Let f = x², so f’ = 2x.
Let g = sin(x), so g’ = cos(x).

Step 2. Combine:
y’ = 2x · sin(x) + x² · cos(x)
Answer: dy/dx = 2x · sin(x) + x² · cos(x)

SAT Math Problems

Problem types that appear on the Digital SAT Math section — heart of algebra, problem solving, and advanced math.

Problem 17 — Slope of a Line

What is the slope of the line passing through (2, 3) and (6, 11)?

Step 1. Use the slope formula: m = (y₂ − y₁) / (x₂ − x₁).

Step 2. Substitute:
m = (11 − 3) / (6 − 2) = 8 / 4 = 2
Answer: slope = 2

Problem 18 — Percent Increase

A store raises the price of a $40 item by 25%. What is the new price?

Step 1. Calculate 25% of $40:
0.25 × 40 = 10

Step 2. Add the increase to the original price:
40 + 10 = 50
Answer: $50

Problem 19 — Mean (Average)

The mean of five numbers is 12. Four of the numbers are 10, 11, 13, and 15. What is the fifth number?

Step 1. The mean tells us the total sum: 12 × 5 = 60.

Step 2. Subtract the known values from the total:
60 − (10 + 11 + 13 + 15) = 60 − 49 = 11
Answer: the fifth number is 11

Problem 20 — Exponent Rules

Simplify: (x³ · x⁵) / x²

Step 1. Multiply same-base exponents by adding: x³ · x⁵ = x⁸.

Step 2. Divide same-base exponents by subtracting: x⁸ / x² = x⁶.
Answer: x⁶

Problem 21 — Linear Equation in Context

A taxi charges $3 plus $2 per mile. Write an equation for the total fare F after m miles, then find F when m = 8.

Step 1. Set up: F = 3 + 2m.

Step 2. Substitute m = 8:
F = 3 + 2(8) = 3 + 16 = 19
Answer: F = $19

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