Question

7. Use the quadratic equation x2−18x=−80

2
−
18

=
−
80
to answer the questions.

Part A: Suppose the equivalent quadratic equation is written x2−18x+c=−80+c

2
−
18

+

=
−
80
+

. 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
(

−
ℎ
)
2
=

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 \( x^2 - 18x = -80 \), let's go through each part step-by-step.

### Part A: Finding c for a Perfect Square Trinomial
The equation can be rewritten as:
\[ x^2 - 18x + c = -80 + c \]

To determine c such that \( x^2 - 18x + c \) is a perfect square trinomial, we need to complete the square.

The general form for completing the square for the equation \( x^2 - bx \) is:
\[ c = \left(\frac{b}{2}\right)^2 \]

Here, \( b = 18 \):
\[ c = \left(\frac{18}{2}\right)^2 = 9^2 = 81 \]

So, the value of c that makes the equation a perfect square trinomial is:
**Part A: c = 81**

### Part B: Factor the Perfect Square Trinomial
Now we can rewrite the equation including our value of c:
\[ x^2 - 18x + 81 = -80 + 81 \]
This simplifies to:
\[ x^2 - 18x + 81 = 1 \]

The left-hand side factors as:
\[ (x - 9)^2 = 1 \]

So we can rewrite the equation as:
**Part B: (x - 9)² = 1**

### Part C: Solve for x (lesser number)
To solve the equation \( (x - 9)^2 = 1 \), we take the square root of both sides:
\[ x - 9 = \pm 1 \]

This leads to two solutions:
1. \( x - 9 = 1 \) → \( x = 10 \)
2. \( x - 9 = -1 \) → \( x = 8 \)

The lesser number from these solutions is:
**Part C: x = 8**

### Part D: Solve for x (greater number)
The greater number from the solutions derived above is:
**Part D: x = 10**

### Summary of Values
- **Part A: c = 81**
- **Part B: (x - 9)² = 1**
- **Part C: x = 8**
- **Part D: x = 10**