Question

using a quadratic formula, find the solution to x^2 - 6x - 16 = 0

Answer 1

To solve the quadratic equation \( x^2 - 6x - 16 = 0 \) using the quadratic formula, we start by identifying the coefficients \(a\), \(b\), and \(c\) from the standard form of a quadratic equation \(ax^2 + bx + c = 0\).

Here, we have:
- \(a = 1\)
- \(b = -6\)
- \(c = -16\)

The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Now, we can substitute the values of \(a\), \(b\), and \(c\) into the formula:

1. First, calculate the discriminant \(b^2 - 4ac\):

\[
b^2 = (-6)^2 = 36
\]
\[
4ac = 4 \times 1 \times (-16) = -64
\]
\[
b^2 - 4ac = 36 - (-64) = 36 + 64 = 100
\]

2. Now substitute \(b\) and the discriminant back into the quadratic formula:

\[
x = \frac{-(-6) \pm \sqrt{100}}{2 \times 1}
\]
\[
x = \frac{6 \pm 10}{2}
\]

3. This will give us two solutions:

For \(x = \frac{6 + 10}{2}\):

\[
x = \frac{16}{2} = 8
\]

For \(x = \frac{6 - 10}{2}\):

\[
x = \frac{-4}{2} = -2
\]

Thus, the solutions to the equation \(x^2 - 6x - 16 = 0\) are:

\[
\boxed{8} \quad \text{and} \quad \boxed{-2}
\]