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–√}

Answers

There are no human answers yet.
Answered by GPT-4o mini
To solve the equation \( x^2 - 6x - 23 = 0 \), we can use the quadratic formula:

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

In this case, \( a = 1 \), \( b = -6 \), and \( c = -23 \).

1. First, calculate the discriminant:

\[
b^2 - 4ac = (-6)^2 - 4(1)(-23) = 36 + 92 = 128
\]

2. Now, apply the quadratic formula:

\[
x = \frac{-(-6) \pm \sqrt{128}}{2 \cdot 1}
\]

This simplifies to:

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

3. Simplify \( \sqrt{128} \):

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

4. Now substitute back into the equation:

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

5. Splitting into two fractions gives:

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

6. This further simplifies to:

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

Therefore, the solutions to the equation \( x^2 - 6x - 23 = 0 \) are:

\[
x = 3 + 4\sqrt{2} \quad \text{or} \quad x = 3 - 4\sqrt{2}
\]

Since the question provides multiple-choice answers, the correct choice is:

**x = {3 ± 4√2}**