How to Solve Quadratic Equations: Complete Guide with Examples
Quadratic Formula, Factoring, Completing the Square & Graphing Methods
Learn how to solve quadratic equations using four methods: quadratic formula, factoring, completing the square, and graphing. Step-by-step examples for each approach.
What You'll Learn
- •All four methods for solving quadratic equations
- •Quadratic formula with step-by-step examples
- •Factoring method and completing the square
- •Discriminant explanation with solution type table
- •Real-world applications in physics and economics
- •Complex number solutions covered
- •Method comparison for choosing the best approach
- •SEO-optimized FAQ section
- •Beginner-friendly progressive difficulty
- •Internal linking to scientific calculator
Full Guide
A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. Quadratic equations are fundamental in algebra and appear in physics, engineering, economics, and many other fields.
The Standard Form
ax² + bx + c = 0
Where:
- a is the coefficient of x² (the quadratic term)
- b is the coefficient of x (the linear term)
- c is the constant term
Method 1: The Quadratic Formula (Always Works)
The quadratic formula solves any quadratic equation:
x = [−b ± √(b² − 4ac)] ÷ 2a
Example:
Solve 2x² − 4x − 6 = 0
a = 2, b = −4, c = −6
x = [4 ± √(16 + 48)] ÷ 4
x = [4 ± √64] ÷ 4
x = [4 ± 8] ÷ 4
x₁ = (4 + 8)/4 = 12/4 = 3
x₂ = (4 − 8)/4 = −4/4 = −1
Solutions: x = 3 or x = −1
Method 2: Factoring (When Possible)
To factor a quadratic, find two numbers that multiply to give ac and add to give b.
Example:
Solve x² + 5x + 6 = 0
Find two numbers multiplying to 6 and adding to 5:
2 × 3 = 6, 2 + 3 = 5 ✓
Factor: (x + 2)(x + 3) = 0
Set each factor to zero:
x + 2 = 0 → x = −2
x + 3 = 0 → x = −3
Solutions: x = −2 or x = −3
Method 3: Completing the Square
Transform the equation into the form (x + p)² = q, then solve.
Example:
Solve x² + 6x − 7 = 0
Step 1: Move constant: x² + 6x = 7
Step 2: Add (b/2)² = (6/2)² = 9 to both sides: x² + 6x + 9 = 16
Step 3: Factor left: (x + 3)² = 16
Step 4: Square root: x + 3 = ±4
x = −3 + 4 = 1
x = −3 − 4 = −7
Solutions: x = 1 or x = −7
Method 4: Graphing
Graph the parabola y = ax² + bx + c. The solutions are the x-coordinates where the parabola crosses the x-axis.
A parabola can:
- Cross the x-axis at TWO points (2 real solutions)
- Touch the x-axis at ONE point (1 real solution — double root)
- NOT cross the x-axis (0 real solutions, 2 complex solutions)
The Discriminant
The discriminant, D = b² − 4ac, tells you about the solutions:
| Discriminant | Solutions |
|---|---|
| D > 0 | Two distinct real solutions |
| D = 0 | One real solution (double root) |
| D < 0 | Two complex (imaginary) solutions |
Example Using the Discriminant:
For x² − 4x + 5 = 0:
D = (−4)² − 4(1)(5) = 16 − 20 = −4
Since D < 0, there are two complex solutions:
x = [4 ± √(−4)] ÷ 2 = [4 ± 2i] ÷ 2 = 2 ± i
Real-World Applications
Physics: Projectile Motion
The height of an object thrown upward: h = −16t² + vt + h₀
Solve for t to find when the object hits the ground.
Economics: Profit Maximization
Profit = −ax² + bx − c
The vertex of the parabola gives the quantity that maximizes profit.
Engineering: Bridge Design
Parabolic arches are described by quadratic equations.
Method Comparison
| Method | When to Use | Difficulty |
|---|---|---|
| Quadratic Formula | Any quadratic, especially with irrational roots | Medium |
| Factoring | Simple integer coefficients | Easy |
| Completing the Square | When coefficient of x² is 1 | Medium |
| Graphing | Visual understanding, checking work | Medium |
FAQ: Quadratic Equations
What is the quadratic formula?
x = [−b ± √(b² − 4ac)] ÷ 2a, where the equation is in the form ax² + bx + c = 0.
What if the coefficient a is negative?
Multiply the entire equation by −1 to make a positive. The quadratic formula still works with negative a.
What is a double root?
When the discriminant is zero, the equation has one repeated solution (one x-intercept).
How do I factor a quadratic when a is not 1?
Use the AC method: multiply a and c, find factors that sum to b, then factor by grouping.
Can all quadratics be factored?
No. Quadratics with irrational or complex roots cannot be factored over real numbers. Use the quadratic formula for these cases.
What is vertex form?
Vertex form: y = a(x − h)² + k, where (h, k) is the vertex. Convert from standard form by completing the square.
What if the quadratic has no constant term (c = 0)?
Factor out x: x(ax + b) = 0. Solutions: x = 0 or x = −b/a.