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Original Question
Solve the equation for all values of x by completing the square. Express your answer in simplest form. x, squared, minus, 8, x,...Question
Solve the equation for all values of x by completing the square. Express your answer in simplest form.
x, squared, plus, 14, x, plus, 45, equals, 0
x
2
+14x+45=
0
x, squared, plus, 14, x, plus, 45, equals, 0
x
2
+14x+45=
0
Answers
Answered by
GPT-4o mini
To solve the equation \( x^2 + 14x + 45 = 0 \) by completing the square, follow these steps:
1. **Move the constant to the other side of the equation:**
\[
x^2 + 14x = -45
\]
2. **Complete the square for the left side:**
Take the coefficient of \( x \), which is \( 14 \), divide it by \( 2 \), and square it:
\[
\left(\frac{14}{2}\right)^2 = 7^2 = 49
\]
Add \( 49 \) to both sides of the equation:
\[
x^2 + 14x + 49 = -45 + 49
\]
Simplifying the right side:
\[
x^2 + 14x + 49 = 4
\]
3. **Rewrite the left side as a squared term:**
The left side can now be factored:
\[
(x + 7)^2 = 4
\]
4. **Take the square root of both sides:**
\[
x + 7 = \pm 2
\]
5. **Solve for \( x \):**
- If \( x + 7 = 2 \):
\[
x = 2 - 7 = -5
\]
- If \( x + 7 = -2 \):
\[
x = -2 - 7 = -9
\]
6. **Final answers:**
The solutions to the equation \( x^2 + 14x + 45 = 0 \) are:
\[
x = -5 \quad \text{and} \quad x = -9
\]
Thus, the values of \( x \) are \( \boxed{-5} \) and \( \boxed{-9} \).
1. **Move the constant to the other side of the equation:**
\[
x^2 + 14x = -45
\]
2. **Complete the square for the left side:**
Take the coefficient of \( x \), which is \( 14 \), divide it by \( 2 \), and square it:
\[
\left(\frac{14}{2}\right)^2 = 7^2 = 49
\]
Add \( 49 \) to both sides of the equation:
\[
x^2 + 14x + 49 = -45 + 49
\]
Simplifying the right side:
\[
x^2 + 14x + 49 = 4
\]
3. **Rewrite the left side as a squared term:**
The left side can now be factored:
\[
(x + 7)^2 = 4
\]
4. **Take the square root of both sides:**
\[
x + 7 = \pm 2
\]
5. **Solve for \( x \):**
- If \( x + 7 = 2 \):
\[
x = 2 - 7 = -5
\]
- If \( x + 7 = -2 \):
\[
x = -2 - 7 = -9
\]
6. **Final answers:**
The solutions to the equation \( x^2 + 14x + 45 = 0 \) are:
\[
x = -5 \quad \text{and} \quad x = -9
\]
Thus, the values of \( x \) are \( \boxed{-5} \) and \( \boxed{-9} \).
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