Factoring Quadratics With Difference of Squares (Math Masters)

What is the factored form of x²-9?

• (x - 9)(x + 9)

• (x - 3)(x + 3)

• (x - 1)(x + 1)

• (x - 3)(x - 3)

Answer explanation

The expression x² - 9 is a difference of squares, which factors as (x - 3)(x + 3). This is because a² - b² = (a - b)(a + b), where a = x and b = 3.

Which of the following expressions is a difference of squares?

• x²+16

• x²-16

• x²+4x+4

• x²+9x

Answer explanation

The expression x² - 16 is a difference of squares because it can be factored as (x - 4)(x + 4). The other options do not fit this form.

What is the factored form of a²-25?

• (a - 5)(a + 5)

• (a - 10)(a + 10)

• (a - 1)(a + 1)

• (a - 25)(a + 25)

Answer explanation

The expression a² - 25 is a difference of squares, which factors as (a - 5)(a + 5). This is because a² - b² = (a - b)(a + b) where b = 5. Thus, the correct choice is (a - 5)(a + 5).

Which of the following is NOT a difference of squares?

• x²-1

• 4y²-16

• x²+4

• 9a²-25

Answer explanation

The expression x² + 4 is not a difference of squares because it cannot be factored into the form (a² - b²). The other options can be expressed as differences of squares: x² - 1 = (x - 1)(x + 1), 4y² - 16 = (2y - 4)(2y + 4), and 9a² - 25 = (3a - 5)(3a + 5).

Factor the expression x²-49.

• (x - 14)(x + 14)

• (x - 49)(x + 1)

• (x - 7)(x - 7)

• (x - 7)(x + 7)

Answer explanation

The expression x² - 49 is a difference of squares, which factors as (x - 7)(x + 7). The other options do not correctly represent this factorization.

What is the factored form of y²-36?

• (y - 6)(y + 6)

• (y - 12)(y + 12)

• (y - 6)(y - 6)

• (y + 6)(y + 6)

Answer explanation

The expression y² - 36 is a difference of squares, which factors as (y - 6)(y + 6). This is because a² - b² = (a - b)(a + b), where a = y and b = 6.

Factor 4x²-25.

• (4x - 5)(x + 5)

• (2x - 10)(2x + 10)

• (2x - 5)(2x + 5)

• (2x - 1)(2x + 1)

Answer explanation

To factor 4x² - 25, recognize it as a difference of squares: (2x)² - 5². This factors to (2x - 5)(2x + 5), which is the correct choice. Other options do not represent the correct factorization.

Which expression can be factored as a difference of squares?

• x²-16y²

• x²+16y²

• x²+8

• x²+4y²

Answer explanation

The expression x² - 16y² can be factored as (x - 4y)(x + 4y), which is a difference of squares. The other options do not fit this form.

What is the factored form of 16-z²?

• (4 - z)(4 + z)

• (8 - z)(2 + z)

• (2 - z)(2 + z)

• (4 - z)(4 - z)

Answer explanation

The expression 16 - z² is a difference of squares, which factors as (a - b)(a + b) where a² = 16 and b² = z². Here, a = 4 and b = z, leading to the factored form (4 - z)(4 + z).

If a rectangle has an area of 64-x², what is the width if the length is 8+x?

• (8+x)²

• 8-x

• 64 / (8 + x)

• 64+x

Answer explanation

The area of a rectangle is given by length times width. Here, area = 64 - x² and length = 8 + x. Thus, width = (64 - x²) / (8 + x). Simplifying gives width = 8 - x, which is the correct answer.