Question

Use the table to identify values of p and q that can be used to factor

x²-x-12 as (x+ p)(x+q).
P 9p+q
2-6 -4
-2 6 4
3-4
-3 4
-1
1

Answer 1

Explanation:

To factor the quadratic expression x² - x - 12 as (x + p)(x + q), we need to find values of p and q that satisfy the following conditions:

1. The coefficient of x² is 1, so p and q should multiply to give 1: p * q = 1.

2. The constant term is -12, so p and q should add up to give -1: p + q = -1.

By considering the given table, we can identify values of p and q that satisfy these conditions:

- For the row "P = 2" and "9p + q = -6", we have p = 2 and q = -4, which satisfies both conditions: 2 * (-4) = -8 and 2 + (-4) = -2.

- For the row "P = -2" and "9p + q = 6", we have p = -2 and q = 4, which also satisfies both conditions: (-2) * 4 = -8 and (-2) + 4 = 2.

- For the row "P = 3" and "9p + q = -4", we have p = 3 and q = -3, which satisfies both conditions: 3 * (-3) = -9 and 3 + (-3) = 0.

- For the row "P = -3" and "9p + q = 4", we have p = -3 and q = 3, which again satisfies both conditions: (-3) * 3 = -9 and (-3) + 3 = 0.

Therefore, the values of p and q that can be used to factor x² - x - 12 as (x + p)(x + q) are:

- p = 2 and q = -4

- p = -2 and q = 4

- p = 3 and q = -3

- p = -3 and q = 3

Answer 2

3 and -4

Explanation:

To factor a quadratic in the form ax² + bx + c, we need to find two numbers that sum to "b" and multiply to the product of "a and c".

For the given quadratic x² - x - 12, the values of a, b and c are:

• a = 1
• b = -1
• c = -12

Therefore, to identity values of p and q that can be used to factor x² - x - 12, we need to find two numbers that sum to -1 and multiply to -12.

The factor pairs of -12 are:

• -1 and 12
• 1 and -12
• -2 and 6
• 2 and -6
• -3 and 4
• 3 and -4

The sum of the factor pairs are:

• -1 + 12 = 11
• 1 + (-12) = -11
• -2 + 6 = 4
• 2 + (-6) = -4
• -3 + 4 = 1
• 3 + (-4) = -1

The only factor pair that sums to -1 is 3 and -4.

Therefore, the pair of numbers that can be used to factor the given quadratic are:

`\large\boxed{\textsf{3 and -4}}`

To prove this, rewrite "b" as the sum of 3 and -4, factor the first two terms and the second two terms separately, then factor out the common term:

`\begin{aligned}x^2 - x - 12&=x^2+3x-4x-12\\&=x(x+3)-4(x+3)\\&=(x+3)(x-4)\end{aligned}`