Solve x2 = 12x – 15 by completing the square. Which is the solution set of the equation?

Question

(negative 6 minus StartRoot 51 EndRoot comma negative 6 + StartRoot 51 EndRoot)
(negative 6 minus StartRoot 21 EndRoot comma negative 6 + StartRoot 21 EndRoot)
(6 minus StartRoot 51 EndRoot comma 6 + StartRoot 51 EndRoot)
(6 minus StartRoot 21 EndRoot comma 6 + StartRoot 21 EndRoot)

Answer 1

To solve the equation \( x^2 = 12x - 15 \) by completing the square, we start by rearranging the equation into a standard form:

\[ x^2 - 12x + 15 = 0 \]

Next, we will complete the square for the quadratic expression \( x^2 - 12x \).

1. Take the coefficient of \( x \), which is \( -12 \), divide it by \( 2 \), and square it: \[ \left(-\frac{12}{2}\right)^2 = (-6)^2 = 36 \]

2. Add and subtract this square (36) inside the equation: \[ x^2 - 12x + 36 - 36 + 15 = 0 \] This simplifies to: \[ (x - 6)^2 - 21 = 0 \]

3. Rearranging gives us: \[ (x - 6)^2 = 21 \]

4. Now, take the square root of both sides: \[ x - 6 = \pm \sqrt{21} \]

5. Solve for \( x \): \[ x = 6 \pm \sqrt{21} \]

Thus, the solutions can be expressed as: \[ x = 6 - \sqrt{21}, \quad x = 6 + \sqrt{21} \]

Therefore, the solution set of the equation is: \[ (6 - \sqrt{21}, 6 + \sqrt{21}) \]

The correct answer is: \((6 - \sqrt{21}, 6 + \sqrt{21})\)