Math Examples & Step-by-Step Solutions
Learn by example with our comprehensive collection of worked math problems. Each example includes detailed step-by-step solutions to help you understand the process.
🌟 Featured Examples
Solving Quadratic Equations
IntermediateProblem:
Solve $2x^2 - 7x + 3 = 0$ using the quadratic formula.
Solution Preview:
Using $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ with $a=2$, $b=-7$, $c=3$...
Finding Triangle Area
BeginnerProblem:
Find the area of a triangle with base 8 cm and height 6 cm.
Solution Preview:
Using the formula $A = \frac{1}{2}bh$ where $b=8$ and $h=6$...
Derivative of Composite Functions
AdvancedProblem:
Find $\frac{d}{dx}[\sin(3x^2 + 1)]$ using the chain rule.
Solution Preview:
Let $u = 3x^2 + 1$, then $\frac{d}{dx}[\sin u] = \cos u \cdot \frac{du}{dx}$...
📚 Examples by Subject
📐 Algebra Examples
Linear Equations
BeginnerProblem: Solve $3x + 7 = 22$
Show Solution
Step 1: Subtract 7 from both sides
$3x = 15$
Step 2: Divide by 3
$x = 5$
Factoring Polynomials
IntermediateProblem: Factor $x^2 - 9x + 20$
Show Solution
Find factors of 20 that add to -9:
$-4 \times -5 = 20$ and $-4 + (-5) = -9$
Therefore: $(x - 4)(x - 5)$
Systems of Equations
IntermediateProblem: Solve the system:
$\begin{cases} 2x + y = 7 \\ x - y = 2 \end{cases}$
Show Solution
Add equations: $3x = 9$, so $x = 3$
Substitute: $2(3) + y = 7$, so $y = 1$
Solution: $(3, 1)$
🔺 Geometry Examples
Pythagorean Theorem
BeginnerProblem: Find the hypotenuse of a right triangle with legs 3 and 4.
Show Solution
Using $c^2 = a^2 + b^2$:
$c^2 = 3^2 + 4^2 = 9 + 16 = 25$
$c = \sqrt{25} = 5$
Circle Area and Circumference
IntermediateProblem: Find area and circumference of a circle with radius 5 cm.
Show Solution
Area: $A = \pi r^2 = \pi(5)^2 = 25\pi$ cm²
Circumference: $C = 2\pi r = 2\pi(5) = 10\pi$ cm
📈 Calculus Examples
Derivative Using Product Rule
AdvancedProblem: Find $\frac{d}{dx}[x^2 \sin x]$
Show Solution
Using product rule: $(uv)' = u'v + uv'$
$u = x^2$, $u' = 2x$
$v = \sin x$, $v' = \cos x$
Result: $2x \sin x + x^2 \cos x$
Finding Limits
AdvancedProblem: Find $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$
Show Solution
Factor numerator: $\frac{(x-2)(x+2)}{x-2}$
Cancel common factor: $x + 2$
Evaluate at $x = 2$: $2 + 2 = 4$
🌊 Trigonometry Examples
Solving Triangles
IntermediateProblem: In triangle ABC, $a = 5$, $b = 7$, $C = 60°$. Find side $c$.
Show Solution
Using Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos C$
$c^2 = 5^2 + 7^2 - 2(5)(7)\cos 60°$
$c^2 = 25 + 49 - 70(0.5) = 39$
$c = \sqrt{39} \approx 6.24$
Trigonometric Identities
IntermediateProblem: Simplify $\frac{\sin^2 x}{1 + \cos x}$
Show Solution
Use identity: $\sin^2 x = 1 - \cos^2 x$
$\frac{1 - \cos^2 x}{1 + \cos x} = \frac{(1-\cos x)(1+\cos x)}{1 + \cos x}$
Cancel: $1 - \cos x$
💡 How to Learn from Examples
1. Read Carefully
Understand what the problem is asking before looking at the solution.
2. Follow Each Step
Don't skip steps. Each one builds on the previous to reach the solution.
3. Practice Similar Problems
Use our calculators and practice problems to reinforce your understanding.
4. Check Your Work
Always verify your answer by substituting back into the original problem.