Quadratic Formula Guide: Solve Any Quadratic Equation Step-by-Step

The quadratic formula is one of the most powerful tools in algebra. It lets you solve any quadratic equation, even when factoring seems impossible. In this guide, you'll learn the formula, understand the discriminant, and see real-world applications using MathAI GPT.

1) What is the Quadratic Formula?

For any quadratic equation in standard form:

The Quadratic Formula

For equation: ax² + bx + c = 0

x = (-b ± √(b² - 4ac)) / 2a

Where:

a = coefficient of x²
b = coefficient of x
c = constant term
± means you get two solutions (one with +, one with −)

2) Step-by-Step Example

Problem: Solve 2x² + 5x - 3 = 0

Step 1: Identify a, b, c

a = 2
b = 5
c = -3

Step 2: Calculate the discriminant

b² - 4ac = 5² - 4(2)(-3) = 25 + 24 = 49

Step 3: Apply the quadratic formula

x = (-5 ± √49) / 4 = (-5 ± 7) / 4

Step 4: Find both solutions

x₁ = (-5 + 7) / 4 = 2 / 4 = 0.5
x₂ = (-5 - 7) / 4 = -12 / 4 = -3

Answer: x = 0.5 or x = -3

3) Understanding the Discriminant

The expression under the square root, b² - 4ac, is called the discriminant. It tells you what kind of solutions to expect:

Discriminant Rules

✓ Positive (> 0): Two different real solutions
✓ Zero (= 0): One repeated real solution (vertex touches x-axis)
✓ Negative (< 0): Two complex (imaginary) solutions, no real solutions

Example: One Solution (Discriminant = 0)

Solve: x² - 6x + 9 = 0

b² - 4ac = (-6)² - 4(1)(9) = 36 - 36 = 0

x = 6 / 2 = 3

Only one solution: x = 3

(This is a perfect square: (x - 3)² = 0)

Example: No Real Solutions (Discriminant < 0)

Solve: x² + 2x + 5 = 0

b² - 4ac = 2² - 4(1)(5) = 4 - 20 = -16

Since the discriminant is negative, there are no real solutions.

The parabola doesn't cross the x-axis. (Complex solutions: x = -1 ± 2i)

4) Finding the Vertex

The vertex (highest or lowest point) of a parabola y = ax² + bx + c is at:

x = -b / 2a

Then substitute this x-value back into the equation to find the y-coordinate.

Example: Find the vertex of y = 2x² + 8x + 3

x = -8 / (2×2) = -8 / 4 = -2

y = 2(-2)² + 8(-2) + 3 = 8 - 16 + 3 = -5

Vertex: (-2, -5)

5) Real-World Applications

Projectile Motion (Physics)

When you throw a ball, its height follows a quadratic path. To find when it hits the ground, solve for when height = 0.

Example: A ball's height is h(t) = -5t² + 20t + 2 meters. When does it hit the ground?

Set h(t) = 0: -5t² + 20t + 2 = 0

Using the quadratic formula:

t = (-20 ± √(400 + 40)) / (-10) ≈ 4.10 seconds

(Taking the positive solution)

Business Optimization

Maximize profit or revenue when it follows a quadratic model. The vertex formula x = -b/(2a) gives the optimal quantity.

Engineering & Design

Parabolic arches, satellite dishes, and bridge cables all use quadratic equations. Finding roots helps determine where structures meet the ground or other surfaces.

6) Common Mistakes to Avoid

⚠️ Watch Out For:

Sign errors: -b means flip the sign of b
Missing the ± symbol: Don't forget both solutions!
Forgetting to multiply 4ac: The discriminant is b² - 4ac, not b² - ac
Dividing only part by 2a: The entire numerator goes over 2a
Using wrong values: Make sure equation is in standard form first

7) When to Use the Quadratic Formula

Always works: Unlike factoring, the quadratic formula solves any quadratic
Non-factorable equations: When numbers don't factor nicely
Quick verification: Check your factoring work
Real-world problems: Physics, engineering, business calculations

8) Practice Problems

Try solving these on your own, then check with our calculator:

x² - 5x + 6 = 0 (Answer: x = 2 or x = 3)
3x² + 7x - 6 = 0 (Answer: x ≈ 0.637 or x ≈ -3.137)
x² + 4x + 4 = 0 (Answer: x = -2)
2x² - x - 15 = 0 (Answer: x = 3 or x = -2.5)
x² + x + 1 = 0 (Answer: No real solutions)

Get Instant Solutions

Need to solve a quadratic equation quickly? Try our Quadratic Formula Calculator for step-by-step solutions with discriminant analysis and vertex coordinates.

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