Question

Solve the following quadratic equation by completing the square.3 x squared minus 42 x plus 135equals0Question content area bottomPart 1xequalsenter your response here​(Simplify your answer. Type an exact​ answer, using radicals as needed. Use a comma to separate answers as​ needed.)

Answer 1

To solve the quadratic equation \(3x^2 - 42x + 135 = 0\) by completing the square, follow these steps:

1. **Move the constant to the other side of the equation:**
\[
3x^2 - 42x = -135
\]

2. **Divide the entire equation by 3 to simplify the leading coefficient of \(x^2\):**
\[
x^2 - 14x = -45
\]

3. **Complete the square:**
- Take half of the coefficient of \(x\) (which is -14), square it, and add to both sides.
- Half of -14 is \(-7\), and \((-7)^2 = 49\).
\[
x^2 - 14x + 49 = -45 + 49
\]
\[
x^2 - 14x + 49 = 4
\]

4. **Rewrite the left-hand side as a squared binomial:**
\[
(x - 7)^2 = 4
\]

5. **Take the square root of both sides:**
\[
x - 7 = \pm 2
\]

6. **Solve for \(x\):**
- For the positive root:
\[
x - 7 = 2 \quad \Rightarrow \quad x = 9
\]
- For the negative root:
\[
x - 7 = -2 \quad \Rightarrow \quad x = 5
\]

So the solutions to the equation \(3x^2 - 42x + 135 = 0\) are:
\[
\boxed{5, 9}
\]