. Use the quadratic equation x2+12x=−20

Question

to answer the questions.

Part A: Suppose the equivalent quadratic equation is written x2+12x+c=−20+c
. What value of c makes the equation a perfect square trinomial?

Part B: Factor the perfect square trinomial, and rewrite the equation as (x−h)2=k

Part C: Solve the quadratic equation and find a solution to x. Enter the lesser number.

Part D: Solve the quadratic equation and find a solution to x. Enter the greater number.

(4 points)
Part A: c =

Part B:

Part C: x =

Part D: x =

Answer 1

To solve the quadratic equation \( x^2 + 12x = -20 \) and answer the questions in parts A through D, we will go step by step.

Part A:

We want to find the value of \( c \) such that \( x^2 + 12x + c \) becomes a perfect square trinomial.

A perfect square trinomial can be expressed in the form \( (x + a)^2 = x^2 + 2ax + a^2 \). Here, \( 2a = 12 \), so:

\[ a = \frac{12}{2} = 6. \]

Therefore, \( c = a^2 = 6^2 = 36 \).

Thus, the value of \( c \) that makes the equation a perfect square trinomial is:

Part A: c = 36

Part B:

Next, we can rewrite the equation as:

\[ x^2 + 12x + 36 = -20 + 36. \]

This simplifies the right side:

\[ x^2 + 12x + 36 = 16. \]

Now, we can factor the left side:

\[ (x + 6)^2 = 16. \]

Thus, our equation in the form \( (x - h)^2 = k \) is:

Part B: (x + 6)² = 16

Part C:

To solve the equation \( (x + 6)^2 = 16 \), we take the square root of both sides:

\[ x + 6 = \pm 4. \]

This provides us with two equations:

1. \( x + 6 = 4 \)
2. \( x + 6 = -4 \)

For \( x + 6 = 4 \):

\[ x = 4 - 6 = -2. \]

For \( x + 6 = -4 \):

\[ x = -4 - 6 = -10. \]

The lesser number is:

Part C: x = -10

Part D:

The greater number is:

Part D: x = -2

Final answers:

• Part A: c = 36
• Part B: (x + 6)² = 16
• Part C: x = -10
• Part D: x = -2