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 a² – b², then a² – b² = (a + b)(a – b)

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

1. Identify if the quadratic expression written in the form of a² – b².
2. Take the square roots of the first and second terms to determine the values of a and b.
3. Apply the formula: a² – b² = (a + b)(a – b)
4. Simplify the factored expression.

Solved Examples

Example 1

Factor x² – 4

x² – 4 can be written in the form a² – b², with a = x and b = 2

Applying the formula:

x²– 4 = (x + 2)(x – 2)

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

Example 2

Factor 9x² – 4

9x² – 4 can be written as (3x)² – 22, with a = 3x and b = 2

Applying the formula:

9x² – 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:

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

Example

Solve x²– 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.

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