Questions: Solve the quadratic equation by completing the square. 36. x^2 + 6x + 2 = 0 37. 9x^2 - 12x = 14

Solution Steps

To solve a quadratic equation by completing the square, follow these steps:

1. Move the constant term to the other side of the equation.
2. Divide the coefficient of the linear term by 2 and square it.
3. Add and subtract this square inside the equation to complete the square.
4. Simplify and solve for \( x \).

Let's apply this approach to the given equations.

1. Move the constant term to the other side: \( x^2 + 6x = -2 \).
2. Divide the coefficient of \( x \) by 2 and square it: \( (6/2)^2 = 9 \).
3. Add and subtract 9: \( x^2 + 6x + 9 = 7 \).
4. Simplify and solve for \( x \).

1. Move the constant term to the other side: \( 9x^2 - 12x = 14 \).
2. Divide the coefficient of \( x \) by 2 and square it: \( (-12/2)^2 = 36 \).
3. Add and subtract 36: \( 9x^2 - 12x + 36 = 50 \).
4. Simplify and solve for \( x \).

For the equation \( x^2 + 6x + 2 = 0 \): \[ x^2 + 6x = -2 \]

For the equation \( 9x^2 - 12x = 14 \): \[ 9x^2 - 12x = 14 \]

For \( x^2 + 6x = -2 \):

1. Divide the coefficient of \( x \) by 2 and square it: \( \left(\frac{6}{2}\right)^2 = 9 \).
2. Add and subtract 9: \[ x^2 + 6x + 9 = 7 \] \[ (x + 3)^2 = 7 \]

For \( 9x^2 - 12x = 14 \):

1. Divide the coefficient of \( x \) by 2 and square it: \( \left(\frac{-12}{2}\right)^2 = 36 \).
2. Add and subtract 36: \[ 9x^2 - 12x + 36 = 50 \] \[ 9(x - \frac{2}{3})^2 = 50 \]

For \( (x + 3)^2 = 7 \): \[ x + 3 = \pm \sqrt{7} \] \[ x = -3 \pm \sqrt{7} \]

For \( 9(x - \frac{2}{3})^2 = 50 \): \[ (x - \frac{2}{3})^2 = \frac{50}{9} \] \[ x - \frac{2}{3} = \pm \sqrt{\frac{50}{9}} \] \[ x = \frac{2}{3} \pm \sqrt{\frac{50}{9}} \] \[ x = \frac{2}{3} \pm \frac{\sqrt{50}}{3} \] \[ x = \frac{2 \pm \sqrt{50}}{3} \]

Final Answer