Question

Solve the equation. Simplify all irrational solutions.

x2 − 6x − 23 = 0


(1 point)
Responses

x = {4±32–√}
x = {4±32–√}

x = {3±42–√}
x = {3±42–√}

x = {±32−−√}
x = {±32−−√}

x = {−3±42–√}

Answer 1

To solve the equation \(x^2 - 6x - 23 = 0\), we can apply the quadratic formula:

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

In this equation:
- \(a = 1\)
- \(b = -6\)
- \(c = -23\)

Now, we can plug these values into the quadratic formula.

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

\[
b^2 = (-6)^2 = 36
\]
\[
4ac = 4 \cdot 1 \cdot (-23) = -92
\]

Now, calculate \(b^2 - 4ac\):

\[
b^2 - 4ac = 36 - (-92) = 36 + 92 = 128
\]

So now we can substitute these values back into the formula:

\[
x = \frac{6 \pm \sqrt{128}}{2}
\]

Next, simplify \(\sqrt{128}\):

\[
\sqrt{128} = \sqrt{64 \cdot 2} = 8\sqrt{2}
\]

Now substituting back:

\[
x = \frac{6 \pm 8\sqrt{2}}{2}
\]

Now simplify this expression:

\[
x = \frac{6}{2} \pm \frac{8\sqrt{2}}{2}
\]
\[
x = 3 \pm 4\sqrt{2}
\]

Finally, we can write the solution as:

\[
x = \{3 \pm 4\sqrt{2}\}
\]

So the correct response from the options given is:

\(x = \{3 \pm 4\sqrt{2}\}\).