Factoring Quadratics: Difference Of Squares Method

Hey guys! Today, we're diving into a super useful technique for factoring quadratic expressions: expressing them as differences of squares. This method can make factoring complex expressions much easier. So, let's get started and break down each problem step by step.

1. Factoring (x² + 4x + 4) - 3²

First, let's focus on the expression (x² + 4x + 4) - 3². Notice that x² + 4x + 4 is a perfect square trinomial. Specifically, it's the square of x + 2. Therefore, we can rewrite the expression as:

(x + 2)² - 3²

Now we have a difference of squares! The difference of squares formula is a² - b² = (a + b)(a - b). Applying this formula, where a = (x + 2) and b = 3, we get:

((x + 2) + 3)((x + 2) - 3)

Simplify each factor:

(x + 5)(x - 1)

So, the factored form of the expression is (x + 5)(x - 1). This method really streamlines the process when you recognize perfect squares.

2. Factoring x² - 4x + 4 - 5²

Next up, we have x² - 4x + 4 - 5². Similar to the first problem, x² - 4x + 4 is a perfect square trinomial. It's the square of (x - 2). So, we rewrite the expression as:

(x - 2)² - 5²

Again, we have a difference of squares. Applying the formula a² - b² = (a + b)(a - b), where a = (x - 2) and b = 5, we get:

((x - 2) + 5)((x - 2) - 5)

Simplify each factor:

(x + 3)(x - 7)

Thus, the factored form is (x + 3)(x - 7). Recognizing these perfect square trinomials saves a lot of effort.

3. Factoring x² + 4x + 3

Now let's tackle x² + 4x + 3. This one isn't immediately a difference of squares, but we can manipulate it to fit that form. We want to complete the square. To do that, we take half of the coefficient of the x term (which is 4), square it (which gives us 4), and add and subtract it within the expression:

x² + 4x + 4 - 4 + 3

Notice we added and subtracted 4, so the value of the expression hasn't changed. Now, we can rewrite the first three terms as a perfect square:

(x + 2)² - 4 + 3

Simplify the constants:

(x + 2)² - 1

Now we have a difference of squares, since 1 is 1². Applying the formula a² - b² = (a + b)(a - b), where a = (x + 2) and b = 1, we get:

((x + 2) + 1)((x + 2) - 1)

Simplify each factor:

(x + 3)(x + 1)

So, the factored form is (x + 3)(x + 1). Completing the square is a powerful technique!

4. Factoring x² - 2x - 3

Let's factor x² - 2x - 3. Again, we'll complete the square. Take half of the coefficient of the x term (which is -2), square it (which gives us 1), and add and subtract it within the expression:

x² - 2x + 1 - 1 - 3

Rewrite the first three terms as a perfect square:

(x - 1)² - 1 - 3

Simplify the constants:

(x - 1)² - 4

Now we have a difference of squares, since 4 is 2². Applying the formula a² - b² = (a + b)(a - b), where a = (x - 1) and b = 2, we get:

((x - 1) + 2)((x - 1) - 2)

Simplify each factor:

(x + 1)(x - 3)

Therefore, the factored form is (x + 1)(x - 3).

5. Factoring 9x² + 6x - 35

Now we have 9x² + 6x - 35. First, notice that 9x² is a perfect square: (3x)². Let's try to complete the square. We can rewrite the expression as:

(9x² + 6x + 1) - 1 - 35

Notice that 9x² + 6x + 1 is (3x + 1)². So we have:

(3x + 1)² - 36

Now we have a difference of squares, since 36 is 6². Applying the formula a² - b² = (a + b)(a - b), where a = (3x + 1) and b = 6, we get:

((3x + 1) + 6)((3x + 1) - 6)

Simplify each factor:

(3x + 7)(3x - 5)

Thus, the factored form is (3x + 7)(3x - 5).

6. Factoring 25x² + 20x - 21

Next, consider 25x² + 20x - 21. We see that 25x² is a perfect square: (5x)². Let's complete the square:

(25x² + 20x + 4) - 4 - 21

Here, 25x² + 20x + 4 is (5x + 2)². So we have:

(5x + 2)² - 25

Now we have a difference of squares, since 25 is 5². Applying the formula a² - b² = (a + b)(a - b), where a = (5x + 2) and b = 5, we get:

((5x + 2) + 5)((5x + 2) - 5)

Simplify each factor:

(5x + 7)(5x - 3)

Therefore, the factored form is (5x + 7)(5x - 3).

7. Factoring x² - 20x + 96

Now let's work on x² - 20x + 96. We complete the square: take half of -20 (which is -10), square it (which is 100), and add and subtract it:

x² - 20x + 100 - 100 + 96

Rewrite the first three terms as a perfect square:

(x - 10)² - 100 + 96

Simplify the constants:

(x - 10)² - 4

Now we have a difference of squares since 4 is 2². Applying the formula a² - b² = (a + b)(a - b), where a = (x - 10) and b = 2, we get:

((x - 10) + 2)((x - 10) - 2)

Simplify:

(x - 8)(x - 12)

Thus, the factored form is (x - 8)(x - 12).

8. Factoring x² - 8x - 9

Let's factor x² - 8x - 9. We complete the square again: Half of -8 is -4, and (-4)² is 16. So we add and subtract 16:

x² - 8x + 16 - 16 - 9

Rewrite the first three terms as a perfect square:

(x - 4)² - 16 - 9

Simplify the constants:

(x - 4)² - 25

Now we have a difference of squares, since 25 is 5². Applying the formula a² - b² = (a + b)(a - b), where a = (x - 4) and b = 5, we get:

((x - 4) + 5)((x - 4) - 5)

Simplify:

(x + 1)(x - 9)

So the factored form is (x + 1)(x - 9).

9. Factoring x² + 20x + 91

Next, we tackle x² + 20x + 91. Completing the square: half of 20 is 10, and 10² is 100. Add and subtract 100:

x² + 20x + 100 - 100 + 91

Rewrite the first three terms as a perfect square:

(x + 10)² - 100 + 91

Simplify the constants:

(x + 10)² - 9

Now we have a difference of squares, since 9 is 3². Applying the formula a² - b² = (a + b)(a - b), where a = (x + 10) and b = 3, we get:

((x + 10) + 3)((x + 10) - 3)

Simplify:

(x + 13)(x + 7)

Thus, the factored form is (x + 13)(x + 7).

10. Factoring x² + x

Finally, let's look at x² + x. This one's a bit different, but we can still use a form of completing the square. First, rewrite the expression as:

x² + x + (1/2)² - (1/2)²

This becomes:

(x + 1/2)² - (1/4)

Now, applying the difference of squares: a² - b² = (a + b)(a - b), where a = (x + 1/2) and b = 1/2, we get:

[(x + 1/2) + 1/2][(x + 1/2) - 1/2]

Simplify:

(x + 1)(x)

So the factored form is x(x + 1). While completing the square works, in this simple case, it's easier to simply factor out an x directly from the original expression: x(x+1)

Conclusion

So, guys, that's how you can factor quadratic expressions by expressing them as differences of squares. Remember to look for perfect square trinomials or complete the square to create the difference of squares pattern. Keep practicing, and you'll get the hang of it in no time! Happy factoring!