Solving Quadratic Equations: Difference of Two Squares

Welcome to this Algebra I lesson on solving quadratic equations using the difference of two squares. This method is essential for simplifying and solving equations that can be expressed in the form \( a^2 - b^2 = 0 \). In this lesson, you will learn the concept, see examples, and practice exercises to reinforce your understanding.

Outline

• Introduction to Difference of Two Squares
• How to Solve Using the Difference of Two Squares
• Examples
• Exercises with Solutions
• Homework
• Revision

Introduction to Difference of Two Squares

The difference of two squares formula is expressed as:

\[ a^2 - b^2 = (a - b)(a + b) \]

To solve quadratic equations using this formula, you need to identify terms that can be written as squares, apply the formula, and solve the resulting linear equations.

How to Solve Using the Difference of Two Squares

Follow these steps to solve an equation using the difference of two squares:

1. Recognize that the quadratic expression is in the form \(a^2 - b^2\).
2. Apply the formula: \(a^2 - b^2 = (a - b)(a + b)\).
3. Set each factor to zero: \(a - b = 0\) and \(a + b = 0\).
4. Solve for the variable.

Examples

Example 1:

Solve the equation: \(x^2 - 16 = 0\).

Solution:

Recognize that \(x^2 - 16\) is a difference of two squares: \((x - 4)(x + 4) = 0\).

Set each factor to zero:

• \(x - 4 = 0 \implies x = 4\)
• \(x + 4 = 0 \implies x = -4\)

The solutions are \(x = 4\) and \(x = -4\).

Exercises with Solutions

Exercise 1:

Solve the equation: \(x^2 - 25 = 0\).

Solution:

Recognize that \(x^2 - 25\) is a difference of two squares: \((x - 5)(x + 5) = 0\).

Set each factor to zero:

• \(x - 5 = 0 \implies x = 5\)
• \(x + 5 = 0 \implies x = -5\)

The solutions are \(x = 5\) and \(x = -5\).

Homework

Solve the following equations using the difference of two squares:

1. \(x^2 - 36 = 0\)
2. \(4x^2 - 49 = 0\)
3. \(9x^2 - 1 = 0\)

Check the solutions in the next revision section.

Revision

Review the examples and exercises to reinforce your understanding of solving quadratic equations using the difference of two squares.

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