Question

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

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

Part B:

Part C:

Part D:

Answer 1

To solve the given quadratic equation \( x^2 - 18x = -80 \), we'll first convert it into the form desired for Part A and then move on through the parts systematically.

### Part A
We want to find a value of \( c \) such that the equation \( x^2 - 18x + c = -80 + c \) becomes a perfect square trinomial.

To complete the square for the expression \( x^2 - 18x \):
1. Take half of the coefficient of \( x \) which is \(-18/2 = -9\).
2. Square that result: \( (-9)^2 = 81 \).

Thus, \( c = 81 \) will make the left side a perfect square trinomial.

### Part B
Now we rewrite the equation adding \( c \):
\[ x^2 - 18x + 81 = -80 + 81 \]
which simplifies to:
\[ x^2 - 18x + 81 = 1 \]

Now, factoring the left-hand side we get:
\[ (x - 9)^2 = 1 \]

### Part C
Now we solve the equation \( (x - 9)^2 = 1 \).

Take the square root of both sides:
1. \( x - 9 = 1 \) or
2. \( x - 9 = -1 \)

Solving these:
1. \( x = 10 \)
2. \( x = 8 \)

The lesser number is \( x = 8 \).

### Part D
The greater number is \( x = 10 \).

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