Question

Solve the equation for all values of x by completing the square.
4, x, squared, plus, 316, equals, 72, x
4x
2
+316=
72x

Answer 1

To solve the equation \(4x^2 + 316 = 72x\) by completing the square, follow these steps:

1. **Rearrange the Equation**:
Start by moving all terms to one side of the equation:

\[
4x^2 - 72x + 316 = 0
\]

2. **Divide by 4**:
To simplify, divide every term by 4:

\[
x^2 - 18x + 79 = 0
\]

3. **Complete the Square**:
To complete the square, take the coefficient of \(x\) (which is -18), divide it by 2, and square it:

\[
\left(-\frac{18}{2}\right)^2 = (-9)^2 = 81
\]

Now, add and subtract 81 inside the equation:

\[
x^2 - 18x + 81 - 81 + 79 = 0
\]
\[
(x - 9)^2 - 2 = 0
\]

4. **Rewrite the Equation**:
Now, rewrite the equation:

\[
(x - 9)^2 = 2
\]

5. **Solve for x**:
Take the square root of both sides:

\[
x - 9 = \pm \sqrt{2}
\]

Now, isolate \(x\):

\[
x = 9 \pm \sqrt{2}
\]

6. **Final Solutions**:
Therefore, the solutions to the equation are:

\[
x = 9 + \sqrt{2} \quad \text{and} \quad x = 9 - \sqrt{2}
\]

So the values of \(x\) are:
\[
x = 9 + \sqrt{2} \quad \text{and} \quad x = 9 - \sqrt{2}
\]