7. 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:

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Answer 1

To solve the quadratic equation \(x^2 - 18x = -80\), we can go through the parts of the problem systematically.

Part A:

To make the equation a perfect square trinomial, we can use the formula for completing the square. A perfect square trinomial takes the form \((x - h)^2 = k\).

1. Start with \(x^2 - 18x\).
2. Take half of the coefficient of \(x\) (which is \(-18\)), divide by \(2\), and square it: \[ \left(\frac{-18}{2}\right)^2 = (-9)^2 = 81 \]
3. Therefore, to make \(x^2 - 18x\) a perfect square, we need to add \(c = 81\) to both sides: \[ x^2 - 18x + 81 = -80 + 81 \] This simplifies to: \[ x^2 - 18x + 81 = 1 \]

Answer Part A:

Value of \(c\): 81

Part B:

Now we can factor the left side: \[ (x - 9)^2 = 1 \]

Answer Part B:

The equation is rewritten as: \((x - 9)^2 = 1\)

Part C:

To solve for \(x\), we take the square root of both sides: \[ x - 9 = \pm 1 \]

This gives us two equations:

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

From the first equation: \[ x = 10 \]

From the second equation: \[ x = 8 \]

Answer Part C:

Lesser number: 8

Part D:

From the values we found in Part C, the greater number is:

Answer Part D:

Greater number: 10