Algebra 5: Quadratic Equations

Learning Objectives

Identify quadratic equations and their standard form
Factor quadratic expressions
Solve by factoring, completing the square, and quadratic formula
Graph quadratic functions and identify key features
Apply quadratics to real-world problems
Understand the relationship between roots and factors

1. Understanding Quadratics

Quadratic Equations

Quadratic equations are equations where the highest power of the variable is 2. They create parabolic curves when graphed.

Standard Form:

ax^2 + bx + c = 0

a = coefficient of x² (cannot be zero)
b = coefficient of x
c = constant term

Examples:

$x^2 + 5x + 6 = 0$
$2x^2 - 3x + 1 = 0$
$x^2 = 9$ (can be written as x² - 9 = 0)

Parabola Explorer

Explore how changing coefficients affects the parabola shape:

Coefficient a: 1

Coefficient b: 0

Constant c: 0

Current equation: y = x² + 0x + 0

Vertex: (-0.0, -0.0)

Axis of Symmetry: x = 0.0

2. Factoring Quadratic Expressions

🔧 Factoring Method

To factor a quadratic means to write it as a product of two binomials.

Example: Factor $x^2 + 5x + 6$

Step 1: Find two numbers that multiply to give 6 and add to give 5

Step 2: The numbers are 2 and 3 (2 × 3 = 6, 2 + 3 = 5)

Step 3: Write as (x + 2)(x + 3)

Check: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6 ✓

🧩 Factoring Practice

Factor the quadratic: x² + 7x + 12

(x + )(x + )

📝 More Factoring Practice

1. x² + 8x + 15 = (x + )(x + )

2. x² - 5x + 6 = (x + )(x + )

3. x² + x - 12 = (x + )(x + )

3. Solving Quadratic Equations

🎯 Zero Product Property

If the product of two factors equals zero, then at least one factor must be zero.

If (x + a)(x + b) = 0, then x + a = 0 or x + b = 0

So: x = -a or x = -b

Example: Solve x² + 5x + 6 = 0

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

Step 2: Set each factor to zero:

x + 2 = 0 → x = -2

x + 3 = 0 → x = -3

Solutions: x = -2, x = -3

Check: (-2)² + 5(-2) + 6 = 4 - 10 + 6 = 0 ✓

(-3)² + 5(-3) + 6 = 9 - 15 + 6 = 0 ✓

🧮 Quadratic Solver

Solve the quadratic equation by factoring:

x² + x + = 0

Factor as: (x + )(x + )

4. The Quadratic Formula

🧮 The Quadratic Formula

The quadratic formula can solve ANY quadratic equation, even if it doesn't factor nicely.

For ax² + bx + c = 0:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The Discriminant (b² - 4ac):

Positive: Two real solutions
Zero: One real solution (perfect square)
Negative: No real solutions (complex numbers)

🧮 Quadratic Formula Calculator

a =

b =

c =

5. Completing the Square

🔲 Completing the Square

Completing the square transforms a quadratic into vertex form: y = a(x - h)² + k

Example: Complete the square for x² + 6x + 8

Step 1: Take half of 6: 6 ÷ 2 = 3

Step 2: Square it: 3² = 9

Step 3: Add and subtract 9: x² + 6x + 9 - 9 + 8

Step 4: Group: (x + 3)² - 1

🔲 Complete the Square

Complete the square for: x² + x +

6. Real-World Applications

🌍 Quadratic Applications

Quadratics appear in many real-world situations involving maximum/minimum values or projectile motion.

🏀 Projectile Motion

A ball thrown upward follows: h = -16t² + 80t

Find maximum height and time to reach it

💰 Revenue Optimization

R = -x² + 100x (price vs quantity)

Find optimal price for maximum revenue

📐 Area Problems

Rectangle with perimeter 20: A = x(10 - x)

Find dimensions for maximum area

🎯 Application Problem

A farmer has 200 feet of fencing to build a rectangular pen. What dimensions give the maximum area?

Area = x × (100 - x) = -x² + 100x

Width: feet

Length: feet

Maximum Area: square feet