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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, 2, x,...Question
Solve the equation for all values of x by completing the square. Express your answer in simplest form.
x, squared, plus, 35, equals, 14, x
x
2
+35=
14x
x, squared, plus, 35, equals, 14, x
x
2
+35=
14x
Answers
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Answered by
GPT-4o mini
To solve the equation \( x^2 + 35 = 14x \) by completing the square, here are the steps:
1. **Rearrange the equation**: Move all terms to one side of the equation:
\[
x^2 - 14x + 35 = 0
\]
2. **Complete the square**: To do this, focus on the \( x^2 - 14x \) part.
- Take half of the coefficient of \( x \) (which is \(-14\)), square it, and add it to both sides. Half of \(-14\) is \(-7\), and squaring it gives \(49\).
- Therefore, we rewrite the equation as follows:
\[
x^2 - 14x + 49 - 49 + 35 = 0
\]
Simplifying this gives:
\[
(x - 7)^2 - 14 = 0
\]
3. **Solve for \( x \)**: Add \( 14 \) to both sides:
\[
(x - 7)^2 = 14
\]
Now, take the square root of both sides:
\[
x - 7 = \pm \sqrt{14}
\]
Thus, we can solve for \( x \):
\[
x = 7 \pm \sqrt{14}
\]
4. **Final solution**:
The solutions for \( x \) in simplified form are:
\[
x = 7 + \sqrt{14} \quad \text{and} \quad x = 7 - \sqrt{14}
\]
These are the values of \( x \) that satisfy the original equation.
1. **Rearrange the equation**: Move all terms to one side of the equation:
\[
x^2 - 14x + 35 = 0
\]
2. **Complete the square**: To do this, focus on the \( x^2 - 14x \) part.
- Take half of the coefficient of \( x \) (which is \(-14\)), square it, and add it to both sides. Half of \(-14\) is \(-7\), and squaring it gives \(49\).
- Therefore, we rewrite the equation as follows:
\[
x^2 - 14x + 49 - 49 + 35 = 0
\]
Simplifying this gives:
\[
(x - 7)^2 - 14 = 0
\]
3. **Solve for \( x \)**: Add \( 14 \) to both sides:
\[
(x - 7)^2 = 14
\]
Now, take the square root of both sides:
\[
x - 7 = \pm \sqrt{14}
\]
Thus, we can solve for \( x \):
\[
x = 7 \pm \sqrt{14}
\]
4. **Final solution**:
The solutions for \( x \) in simplified form are:
\[
x = 7 + \sqrt{14} \quad \text{and} \quad x = 7 - \sqrt{14}
\]
These are the values of \( x \) that satisfy the original equation.
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