Question

Solve the equation for all values of x by completing the square.
3, x, squared, minus, 60, x, equals, minus, 249
3x
2
−60x=
−249

Answer 1

To solve the equation \( 3x^2 - 60x = -249 \) by completing the square, we can follow these steps:

1. **Move all terms to one side:**
\[
3x^2 - 60x + 249 = 0
\]

2. **Divide the entire equation by 3 to simplify:**
\[
x^2 - 20x + 83 = 0
\]

3. **Move the constant term to the other side:**
\[
x^2 - 20x = -83
\]

4. **Complete the square:**
- Take the coefficient of \( x \) (which is -20), divide by 2, and square it.
\[
\left(-\frac{20}{2}\right)^2 = (-10)^2 = 100
\]
- Add 100 to both sides:
\[
x^2 - 20x + 100 = -83 + 100
\]
This simplifies to:
\[
x^2 - 20x + 100 = 17
\]

5. **Rewrite the left-hand side as a square:**
\[
(x - 10)^2 = 17
\]

6. **Take the square root of both sides:**
\[
x - 10 = \pm \sqrt{17}
\]

7. **Solve for \( x \):**
\[
x = 10 \pm \sqrt{17}
\]

Thus, the solutions for \( x \) are:
\[
x = 10 + \sqrt{17} \quad \text{and} \quad x = 10 - \sqrt{17}
\]