Question

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.

Answer 1

To solve the problem, let's start by rewriting and analyzing the given quadratic equation.

The given quadratic equation is:

\[ x^2 - 18x = -80 \]

### Part A: Finding the value of c

To convert the expression \( x^2 - 18x + c = -80 + c \) into a perfect square trinomial, we can complete the square. The term \( c \) can be determined using the formula:

\[
c = \left( \frac{b}{2} \right)^2
\]

where \( b \) is the coefficient of \( x \) (in this case, \( -18 \)).

Calculating \( c \):

\[
b = -18 \implies \frac{b}{2} = \frac{-18}{2} = -9
\]
\[
c = (-9)^2 = 81
\]

Thus, the value of \( c \) that makes the equation a perfect square trinomial is **81**.

### Part B: Factor the perfect square trinomial

Insert the value of \( c \) back into the equation:

\[
x^2 - 18x + 81 = -80 + 81
\]
\[
x^2 - 18x + 81 = 1
\]

The left side can be factored as:

\[
(x - 9)^2 = 1
\]

### Part C: Solve the quadratic equation and find the lesser number

Now we need to solve \( (x - 9)^2 = 1 \). Taking the square root of both sides:

\[
x - 9 = \pm 1
\]

This leads to two equations:

1. \( x - 9 = 1 \)
2. \( x - 9 = -1 \)

Solving each equation gives:

1. \( x = 10 \)
2. \( x = 8 \)

Thus, the lesser number is **8**.

### Part D: Solve the quadratic equation and find the greater number

From the solutions above, the greater number is **10**.

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