How can I solve the quadratic equation 3x² – 18x + 15 = 0 by completing the square?

To solve the quadratic equation 3x² – 18x + 15 = 0 by completing the square, we’ll follow several clear steps. Here’s how:

1. Divide the entire equation by the coefficient of x²: Since the coefficient of x² is 3, we divide every term by 3 to simplify the equation.

2. Rewrite the equation:

    x² - 6x + 5 = 0

3. Rearrange the equation: Move the constant term to the right side of the equation.

4. This gives us:

   x² - 6x = -5

5. Complete the square: To do this, take the coefficient of x (which is -6), halve it to get -3, and then square it to obtain 9.

6. Add and subtract this square on the left side:

   x² - 6x + 9 - 9 = -5

7. Now rewrite the left side as a squared binomial: This simplifies to:

8. (x - 3)² - 9 = -5

9. Add 9 to both sides:

   This leads us to:

   (x - 3)² = 4

10. Take the square root of both sides: Remember to consider both the positive and negative roots:

11. x - 3 = ±2

12. Now solve for x: This gives us two equations:

13. x - 3 = 2  → x = 5

    x - 3 = -2 → x = 1

Conclusion: The solutions to the equation 3x² – 18x + 15 = 0 by completing the square are x = 5 and x = 1.