Question
7. Use the quadratic equation x2−18x=−80
to answer the questions.
Desmos Calculator (click here)
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)
to answer the questions.
Desmos Calculator (click here)
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)
Answers
GPT-4o mini
To solve the quadratic equation \( x^2 - 18x = -80 \), let's rewrite it in a more standard form and work through each part step by step.
### Part A: Find \( c \) for a perfect square trinomial
We start with the form you provided:
\[
x^2 - 18x + c = -80 + c
\]
To make \( x^2 - 18x + c \) a perfect square trinomial, we need to complete the square. The first step is to take half of the coefficient of \( x \) (which is \(-18\)), square it, and add it to both sides:
\[
\left(\frac{-18}{2}\right)^2 = (-9)^2 = 81
\]
Thus, \( c = 81 \). The equation now looks like this:
\[
x^2 - 18x + 81 = -80 + 81
\]
\[
x^2 - 18x + 81 = 1
\]
### Part B: Factor the perfect square trinomial
The left-hand side can be factored as:
\[
(x - 9)^2 = 1
\]
So we rewrite the equation as:
\[
(x - 9)^2 = 1
\]
### Part C: Solve the quadratic equation and find the lesser solution
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. \( x - 9 = -1 \)
Solving these:
1. For \( x - 9 = 1 \):
\[
x = 1 + 9 = 10
\]
2. For \( x - 9 = -1 \):
\[
x = -1 + 9 = 8
\]
The lesser solution is \( x = 8 \).
### Part D: Solve the quadratic equation and find the greater solution
From the previous parts, we found the two solutions:
- \( x = 10 \) (greater solution)
- \( x = 8 \) (lesser solution)
Therefore, the greater solution is \( x = 10 \).
### Summary of Answers:
- **Part A**: \( c = 81 \)
- **Part B**: \( (x - 9)^2 = 1 \)
- **Part C**: Lesser solution \( x = 8 \)
- **Part D**: Greater solution \( x = 10 \)
### Part A: Find \( c \) for a perfect square trinomial
We start with the form you provided:
\[
x^2 - 18x + c = -80 + c
\]
To make \( x^2 - 18x + c \) a perfect square trinomial, we need to complete the square. The first step is to take half of the coefficient of \( x \) (which is \(-18\)), square it, and add it to both sides:
\[
\left(\frac{-18}{2}\right)^2 = (-9)^2 = 81
\]
Thus, \( c = 81 \). The equation now looks like this:
\[
x^2 - 18x + 81 = -80 + 81
\]
\[
x^2 - 18x + 81 = 1
\]
### Part B: Factor the perfect square trinomial
The left-hand side can be factored as:
\[
(x - 9)^2 = 1
\]
So we rewrite the equation as:
\[
(x - 9)^2 = 1
\]
### Part C: Solve the quadratic equation and find the lesser solution
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. \( x - 9 = -1 \)
Solving these:
1. For \( x - 9 = 1 \):
\[
x = 1 + 9 = 10
\]
2. For \( x - 9 = -1 \):
\[
x = -1 + 9 = 8
\]
The lesser solution is \( x = 8 \).
### Part D: Solve the quadratic equation and find the greater solution
From the previous parts, we found the two solutions:
- \( x = 10 \) (greater solution)
- \( x = 8 \) (lesser solution)
Therefore, the greater solution is \( x = 10 \).
### Summary of Answers:
- **Part A**: \( c = 81 \)
- **Part B**: \( (x - 9)^2 = 1 \)
- **Part C**: Lesser solution \( x = 8 \)
- **Part D**: Greater solution \( x = 10 \)