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