Solving Quadratic Equations

What is a Quadratic Equation?

A quadratic equation has the form: ax² + bx + c = 0

Where:

• a ≠ 0 (must have x² term)
• b can be any number (coefficient of x)
• c can be any number (constant)

Examples:

• x² + 5x + 6 = 0
• 2x² - 8 = 0
• x² - 4x = 0

Standard form: ax² + bx + c = 0

Zero Product Property

Key principle: If ab = 0, then a = 0 or b = 0

Used for: Solving factored equations

Example: Basic Application

(x - 3)(x + 2) = 0

Apply zero product property:

• x - 3 = 0 OR x + 2 = 0
• x = 3 OR x = -2

Solutions: x = 3 or x = -2

Check x = 3: (3-3)(3+2) = 0(5) = 0 ✓ Check x = -2: (-2-3)(-2+2) = (-5)(0) = 0 ✓

Solving by Factoring

Method: Factor, then use zero product property

Steps:

1. Write in standard form (= 0)
2. Factor the left side
3. Set each factor equal to zero
4. Solve for x

Example 1: Factor and Solve

x² + 7x + 12 = 0

Step 1: Already in standard form

Step 2: Factor

• Need two numbers that multiply to 12 and add to 7
• Numbers: 3 and 4
• (x + 3)(x + 4) = 0

Step 3: Set each factor to zero

• x + 3 = 0 → x = -3
• x + 4 = 0 → x = -4

Solutions: x = -3 or x = -4

Example 2: Leading Coefficient ≠ 1

2x² + 7x + 3 = 0

Factor: (2x + 1)(x + 3) = 0

Solve:

• 2x + 1 = 0 → x = -1/2
• x + 3 = 0 → x = -3

Solutions: x = -1/2 or x = -3

Example 3: Difference of Squares

x² - 25 = 0

Factor: (x + 5)(x - 5) = 0

Solve:

• x + 5 = 0 → x = -5
• x - 5 = 0 → x = 5

Solutions: x = -5 or x = 5

Example 4: Common Factor First

2x² + 8x = 0

Factor out GCF: 2x(x + 4) = 0

Solve:

• 2x = 0 → x = 0
• x + 4 = 0 → x = -4

Solutions: x = 0 or x = -4

Important: Don't divide by x! You'll lose the solution x = 0.

Solving Using Square Roots

For equations in form: x² = k

Solution: x = ±√k

Remember: Both positive and negative roots!

Example 1: Basic Square Root

x² = 16

Take square root of both sides:

• x = ±√16
• x = ±4

Solutions: x = 4 or x = -4

Example 2: Isolate x² First

x² - 9 = 0

Step 1: Add 9 to both sides

• x² = 9

Step 2: Take square root

• x = ±3

Solutions: x = 3 or x = -3

Example 3: Coefficient on x²

3x² = 75

Step 1: Divide by 3

• x² = 25

Step 2: Take square root

• x = ±5

Solutions: x = 5 or x = -5

Example 4: Non-Perfect Square

x² = 20

Take square root:

• x = ±√20
• x = ±2√5

Solutions: x = 2√5 or x = -2√5

Solving (x - h)² = k

For perfect square form: (x - h)² = k

Solution: x - h = ±√k

Example 1: Shifted Square

(x - 3)² = 16

Take square root:

• x - 3 = ±4

Solve:

• x - 3 = 4 → x = 7
• x - 3 = -4 → x = -1

Solutions: x = 7 or x = -1

Example 2: With Addition

(x + 2)² = 9

Take square root:

• x + 2 = ±3

Solve:

• x + 2 = 3 → x = 1
• x + 2 = -3 → x = -5

Solutions: x = 1 or x = -5

No Real Solutions

If x² = negative number: No real solutions

Example: No Real Solution

x² = -4

Taking square root would give: x = ±√(-4)

No real number squared equals negative!

Answer: No real solutions

Same for: x² + 4 = 0

• x² = -4
• No real solutions

Number of Solutions

Quadratic equations have:

• Two solutions (most common)
• One solution (perfect square, like (x-3)² = 0)
• No real solutions (x² = negative)

Example: One Solution

(x - 5)² = 0

Take square root:

• x - 5 = 0
• x = 5

Only one solution: x = 5

Quadratic Formula (Introduction)

For any quadratic: ax² + bx + c = 0

Quadratic formula: x = [-b ± √(b² - 4ac)] / (2a)

When to use:

• Can't factor easily
• Want exact solutions

Example: Using Formula

x² + 5x + 3 = 0

Identify: a = 1, b = 5, c = 3

Substitute:

x = [-5 ± √(25 - 12)] / 2
x = [-5 ± √13] / 2

Solutions: x = (-5 + √13)/2 or x = (-5 - √13)/2

Applications

Projectile motion: Height over time

• h = -16t² + 64t
• When does object hit ground? (h = 0)

Area problems: Find dimensions

• Rectangle: length × width = 24
• If length = width + 5, find dimensions

Profit maximization: Business applications

• Find price that maximizes profit

Practice