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

Question

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 equation \( x^2 - 18x = -80 \) and answer the parts step by step:

Part A:

To rewrite the equation as a perfect square trinomial, we want to complete the square.

1. Start with the expression \( x^2 - 18x \).
2. Take half of the coefficient of \( x \) (which is -18), square it, and add it to both sides. \[ \left(\frac{-18}{2}\right)^2 = 9^2 = 81 \] Therefore, we need \( c = 81 \).

The equivalent equation is: \[ x^2 - 18x + 81 = -80 + 81 \] This simplifies to: \[ x^2 - 18x + 81 = 1 \]

Part B:

Now, we can factor the left-hand side as a perfect square: \[ (x - 9)^2 = 1 \]

Part C:

To solve for \( x \) in the equation \( (x - 9)^2 = 1 \), we take the square root of both sides: \[ x - 9 = \pm 1 \] Solving these gives:

1. \( x - 9 = 1 \) ⟹ \( x = 10 \)
2. \( x - 9 = -1 \) ⟹ \( x = 8 \)

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

Part D:

The greater solution is \( x = 10 \).

Summary of Answers:

• Part A: 81
• Part B: \( (x - 9)^2 = 1 \)
• Part C: 8
• Part D: 10