x^2- 18x = -80

Question

A. Suppose the equivalent equation is written x^2 -18x +c= -80+ c. What value of c makes the equation a perfect square trinomial
B. Factor the perfect square trinomial and rewrite the equation as (x - h)^2=k
C. Solve the equation and find a solution to x enter the lesser number
D. Solve the equation and find a solution to x enter the greater number

Answer 1

To solve the problem step by step, we start with the equation:

Step A: Determine the value of \(c\) that makes the equation a perfect square trinomial.

We are given the equation:

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

We want to rewrite it in the form \(x^2 - 18x + c = -80 + c\).

To complete the square, we need to take half of the coefficient of \(x\) (which is \(-18\)), square it, and add it to both sides:

1. Half of \(-18\) is \(-9\).
2. Squaring \(-9\) gives \(81\).

Thus, \(c = 81\).

Step B: Factor the perfect square trinomial.

Now we rewrite the equation with \(c\):

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

This simplifies to:

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

Now we can factor the left side:

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

Step C: Solve the equation for the lesser solution.

To find the solutions for \(x\), we take the square root of both sides:

\[ x - 9 = \pm 1 \]

This gives us two equations:

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

Solving these:

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

The lesser solution is:

\[ \text{Lesser solution: } x = 8 \]

Step D: Solve for the greater solution.

The greater solution is:

\[ \text{Greater solution: } x = 10 \]

Final Answers:

A. \(c = 81\)
B. The equation is \((x - 9)^2 = 1\)
C. Lesser number solution: \(x = 8\)
D. Greater number solution: \(x = 10\)