x2−6x−20=7𝑥2−6𝑥−20=7 Type the answer in the box below.x=𝑥=   or   x=𝑥=  Show your reasoning.

1. Break Down the Problem

We need to solve the quadratic equation: $x^2 - 6x - 20 = 7$

2. Relevant Concepts

To solve for $x$, we first need to set the equation to zero by moving all terms to one side.

3. Analysis and Detail

1. Rearranging the equation: $x^2 - 6x - 20 - 7 = 0$ $x^2 - 6x - 27 = 0$

2. Now, we will use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 1$, $b = -6$, and $c = -27$.

3. Calculate the discriminant: $b^2 - 4ac = (-6)^2 - 4(1)(-27)$ $= 36 + 108 = 144$

4. Calculating $x$: $x = \frac{-(-6) \pm \sqrt{144}}{2(1)}$ $= \frac{6 \pm 12}{2}$

5. This results in two potential solutions: $x = \frac{6 + 12}{2} = \frac{18}{2} = 9$ $x = \frac{6 - 12}{2} = \frac{-6}{2} = -3$

4. Verify and Summarize

The solutions for the equation $x^2 - 6x - 27 = 0$ are $x = 9$ and $x = -3$.

Final Answer

The solutions are: $x = 9 \quad \text{or} \quad x = -3$