Question
solve x^2-14x-6=9 by completing the square enter the lesser value first
Answers
GPT-4o mini
To solve the equation \( x^2 - 14x - 6 = 9 \) by completing the square, we first need to move all terms to one side of the equation:
\[
x^2 - 14x - 6 - 9 = 0
\]
\[
x^2 - 14x - 15 = 0
\]
Next, we rearrange it to isolate the constant term:
\[
x^2 - 14x = 15
\]
Now, we complete the square on the left side. To do this, we take the coefficient of \( x \), which is \(-14\), divide it by 2 to get \(-7\), and then square it to get \(49\):
\[
x^2 - 14x + 49 = 15 + 49
\]
\[
x^2 - 14x + 49 = 64
\]
Now, we can write the left side as a square:
\[
(x - 7)^2 = 64
\]
Next, we take the square root of both sides:
\[
x - 7 = \pm 8
\]
This gives us two equations to solve:
1. \( x - 7 = 8 \)
2. \( x - 7 = -8 \)
Solving these equations:
1. For \( x - 7 = 8 \):
\[
x = 8 + 7 = 15
\]
2. For \( x - 7 = -8 \):
\[
x = -8 + 7 = -1
\]
Thus, the solutions are \( x = 15 \) and \( x = -1 \).
To enter the lesser value first, the final answer is:
\[
-1, 15
\]
\[
x^2 - 14x - 6 - 9 = 0
\]
\[
x^2 - 14x - 15 = 0
\]
Next, we rearrange it to isolate the constant term:
\[
x^2 - 14x = 15
\]
Now, we complete the square on the left side. To do this, we take the coefficient of \( x \), which is \(-14\), divide it by 2 to get \(-7\), and then square it to get \(49\):
\[
x^2 - 14x + 49 = 15 + 49
\]
\[
x^2 - 14x + 49 = 64
\]
Now, we can write the left side as a square:
\[
(x - 7)^2 = 64
\]
Next, we take the square root of both sides:
\[
x - 7 = \pm 8
\]
This gives us two equations to solve:
1. \( x - 7 = 8 \)
2. \( x - 7 = -8 \)
Solving these equations:
1. For \( x - 7 = 8 \):
\[
x = 8 + 7 = 15
\]
2. For \( x - 7 = -8 \):
\[
x = -8 + 7 = -1
\]
Thus, the solutions are \( x = 15 \) and \( x = -1 \).
To enter the lesser value first, the final answer is:
\[
-1, 15
\]