# Factoring Quadratics: Difference of Squares

**Factoring** quadratic expressions often be simplified by using the difference of squares formula. This technique is useful when the quadratic expression written as the difference of two perfect squares.

**How to Apply the Difference of Squares Formula**

Difference of squares formula states that the difference between two squares is equal to the product of the sum and difference of their roots.

If a^{2} – b^{2}, then a^{2} – b^{2} = (a + b)(a – b)

To factor a quadratic expression using the difference of squares formula:

- Identify if the quadratic expression written in the form of a
^{2}– b^{2}. - Take the square roots of the first and second terms to determine the values of a and b.
- Apply the formula: a
^{2}– b^{2}= (a + b)(a – b) - Simplify the factored expression.

**Solved Examples**

**Example 1**

Factor x^{2} – 4

x^{2} – 4 can be written in the form a^{2} – b^{2}, with a = x and b = 2

Applying the formula:

x^{2}– 4 = (x + 2)(x – 2)

The factored expression is (x + 2)(x – 2).

**Example 2**

Factor 9x^{2} – 4

9x^{2} – 4 can be written as (3x)^{2} – 22, with a = 3x and b = 2

Applying the formula:

9x^{2} – 4 = (3x + 2)(3x – 2)

The factored expression is (3x + 2)(3x – 2).

**Solving Quadratic Equations Using Factored Form**

Once a quadratic expression has been factored using the difference of squares formula, the factored form can be used to solve quadratic equations.

To solve:

- Set the factored expression equal to 0.
- Solve each factor individually by setting it equal to 0 and solving.
- The solutions will be the values obtained from step 2.

**Example**

Solve x^{2}– 4 = 0

Factoring using difference of squares gives: (x + 2)(x – 2) = 0

Setting each factor equal to 0:

x + 2 = 0, x = -2

x – 2 = 0, x = 2

The solutions are x = -2 and x = 2.

**FAQs**

### What is the difference of squares formula?

The difference of squares formula states that the difference between two perfect squares (a^{2} – b^{2}) can be factored as the product (a + b)(a – b).

### When can you use the difference of squares formula?

You can use it when the quadratic expression can be written in the form of a^{2} – b^{2} for some values of a and b.

### Why does the difference of squares formula work?

It works because the product of (a + b)(a – b) will expand to a^{2} – b^{2}, allowing you to simplify the original quadratic expression.

### What are some examples of factoring using difference of squares?

x^{2} – 4 = (x + 2)(x – 2)

9x^{2}– 1 = (3x + 1)(3x – 1)

4x^{2} – 9 = (2x + 3)(2x – 3)

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