Question

x=\frac{-7\ \pm\sqrt{\left(7\right)^{2}-4\left(2\right)\left(-15\right)}}{2\left(2\right)}

Answer 1

To solve the equation

\[
x = \frac{-7 \pm \sqrt{(7)^2 - 4(2)(-15)}}{2(2)},
\]

let's break it down step-by-step.

1. **Calculate the discriminant**:

\[
(7)^2 - 4(2)(-15) = 49 - 4 \cdot 2 \cdot -15.
\]

Calculating \(4 \cdot 2 \cdot -15\):

\[
4 \cdot 2 = 8,
\]
and
\[
8 \cdot -15 = -120.
\]
Therefore, the discriminant becomes:

\[
49 - (-120) = 49 + 120 = 169.
\]

2. **Take the square root of the discriminant**:

\[
\sqrt{169} = 13.
\]

3. **Substitute back into the formula**:

Now we can substitute back into the formula for \(x\):

\[
x = \frac{-7 \pm 13}{2(2)} = \frac{-7 \pm 13}{4}.
\]

4. **Calculate the two possible values for \(x\)**:

1. For the positive case:

\[
x = \frac{-7 + 13}{4} = \frac{6}{4} = \frac{3}{2}.
\]

2. For the negative case:

\[
x = \frac{-7 - 13}{4} = \frac{-20}{4} = -5.
\]

5. **Final solutions**:

Thus, the solutions for \(x\) are:

\[
x = \frac{3}{2} \quad \text{and} \quad x = -5.
\]

In conclusion, the values of \(x\) are:

\[
\boxed{\frac{3}{2}} \quad \text{and} \quad \boxed{-5}.
\]