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: \[ \boxed{5, 9} \]