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.
(4 points)
Part A: c=
Part B:
Part C: x=
Part D: x=
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: c=
Part B:
Part C: x=
Part D: x=
Answers
GPT-4o mini
To solve the quadratic equation \( x^2 - 18x = -80 \), we will first manipulate it and address each part step by step.
**Part A: Find \( c \) that makes the equation a perfect square trinomial.**
The original equation can be rewritten to find the value of \( c \):
\[ x^2 - 18x + c = -80 + c \]
To complete the square, we take half of the coefficient of \( x \), which is \(-18\), divide it by 2 (which gives \(-9\)), and then square it:
\[ (-9)^2 = 81 \]
Thus, \( c = 81 \) makes the expression a perfect square trinomial.
**Answer for Part A:**
\[ c = 81 \]
---
**Part B: Factor the perfect square trinomial and rewrite the equation as \( (x-h)^2 = k \).**
Now, substituting \( c \) into the equation:
\[ x^2 - 18x + 81 = -80 + 81 \]
\[ x^2 - 18x + 81 = 1 \]
This can be factored as:
\[ (x - 9)^2 = 1 \]
**Answer for Part B:**
\[ (x - 9)^2 = 1 \]
---
**Part C: Solve the quadratic equation and find the lesser solution for \( x \).**
To solve the equation \( (x - 9)^2 = 1 \), we take the square root of both sides:
\[ x - 9 = \pm 1 \]
This gives us two equations:
1. \( x - 9 = 1 \) leads to \( x = 10 \)
2. \( x - 9 = -1 \) leads to \( x = 8 \)
The lesser solution is:
**Answer for Part C:**
\[ x = 8 \]
---
**Part D: Solve the quadratic equation and find the greater solution for \( x \).**
From the above calculations, the greater solution is:
**Answer for Part D:**
\[ x = 10 \]
---
**Final Summary of Answers:**
Part A: \( c = 81 \)
Part B: \( (x - 9)^2 = 1 \)
Part C: \( x = 8 \)
Part D: \( x = 10 \)
**Part A: Find \( c \) that makes the equation a perfect square trinomial.**
The original equation can be rewritten to find the value of \( c \):
\[ x^2 - 18x + c = -80 + c \]
To complete the square, we take half of the coefficient of \( x \), which is \(-18\), divide it by 2 (which gives \(-9\)), and then square it:
\[ (-9)^2 = 81 \]
Thus, \( c = 81 \) makes the expression a perfect square trinomial.
**Answer for Part A:**
\[ c = 81 \]
---
**Part B: Factor the perfect square trinomial and rewrite the equation as \( (x-h)^2 = k \).**
Now, substituting \( c \) into the equation:
\[ x^2 - 18x + 81 = -80 + 81 \]
\[ x^2 - 18x + 81 = 1 \]
This can be factored as:
\[ (x - 9)^2 = 1 \]
**Answer for Part B:**
\[ (x - 9)^2 = 1 \]
---
**Part C: Solve the quadratic equation and find the lesser solution for \( x \).**
To solve the equation \( (x - 9)^2 = 1 \), we take the square root of both sides:
\[ x - 9 = \pm 1 \]
This gives us two equations:
1. \( x - 9 = 1 \) leads to \( x = 10 \)
2. \( x - 9 = -1 \) leads to \( x = 8 \)
The lesser solution is:
**Answer for Part C:**
\[ x = 8 \]
---
**Part D: Solve the quadratic equation and find the greater solution for \( x \).**
From the above calculations, the greater solution is:
**Answer for Part D:**
\[ x = 10 \]
---
**Final Summary of Answers:**
Part A: \( c = 81 \)
Part B: \( (x - 9)^2 = 1 \)
Part C: \( x = 8 \)
Part D: \( x = 10 \)