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To find the values of [tex]\( x \)[/tex] that make the equation [tex]\(-x^2 + 8x = -15\)[/tex] true, we will follow these steps:
1. Rewrite the equation in standard quadratic form: Add 15 to both sides to set the equation to zero.
[tex]\[ -x^2 + 8x + 15 = 0 \][/tex]
2. Identify the coefficients: The quadratic equation is now in the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a = -1 \)[/tex], [tex]\( b = 8 \)[/tex], and [tex]\( c = 15 \)[/tex].
3. Solve the quadratic equation using the quadratic formula, which is given by
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here [tex]\( a = -1 \)[/tex], [tex]\( b = 8 \)[/tex], and [tex]\( c = 15 \)[/tex].
4. Substitute the coefficients into the quadratic formula:
[tex]\[ x = \frac{-8 \pm \sqrt{8^2 - 4(-1)(15)}}{2(-1)} \][/tex]
5. Simplify the terms under the square root:
[tex]\[ x = \frac{-8 \pm \sqrt{64 + 60}}{-2} \][/tex]
[tex]\[ x = \frac{-8 \pm \sqrt{124}}{-2} \][/tex]
6. Simplify the square root and the expression:
[tex]\[ \sqrt{124} \text{ can be written as } \sqrt{4 \cdot 31} = 2\sqrt{31} \][/tex]
[tex]\[ x = \frac{-8 \pm 2\sqrt{31}}{-2} \][/tex]
7. Factor out the common term in the numerator:
[tex]\[ x = \frac{-8 \pm 2\sqrt{31}}{-2} = \frac{-8}{-2} \pm \frac{2\sqrt{31}}{-2} \][/tex]
Simplify both fractions:
[tex]\[ x = 4 \mp \sqrt{31} \][/tex]
Thus, the solutions are:
[tex]\[ x = 4 - \sqrt{31} \quad \text{and} \quad x = 4 + \sqrt{31} \][/tex]
Now, let's match these solutions to the given options. The correct answer is:
A. [tex]\( 4 \pm \sqrt{62} \)[/tex]
B. [tex]\( 4 \pm \sqrt{3 I} \)[/tex]
C. [tex]\( -4 \pm \sqrt{62} \)[/tex]
D. [tex]\( -1 \pm \sqrt{31} \)[/tex]
None of the options match our solutions [tex]\(4 \pm \sqrt{31}\)[/tex], which suggests there may be an error in the provided answer choices. Since we know our calculations are correct, it's possible the correct answer isn't listed.
However, based on the solution we obtained, the final values for [tex]\( x \)[/tex] are:
[tex]\[ 4 - \sqrt{31} \quad \text{and} \quad 4 + \sqrt{31} \][/tex]
1. Rewrite the equation in standard quadratic form: Add 15 to both sides to set the equation to zero.
[tex]\[ -x^2 + 8x + 15 = 0 \][/tex]
2. Identify the coefficients: The quadratic equation is now in the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a = -1 \)[/tex], [tex]\( b = 8 \)[/tex], and [tex]\( c = 15 \)[/tex].
3. Solve the quadratic equation using the quadratic formula, which is given by
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here [tex]\( a = -1 \)[/tex], [tex]\( b = 8 \)[/tex], and [tex]\( c = 15 \)[/tex].
4. Substitute the coefficients into the quadratic formula:
[tex]\[ x = \frac{-8 \pm \sqrt{8^2 - 4(-1)(15)}}{2(-1)} \][/tex]
5. Simplify the terms under the square root:
[tex]\[ x = \frac{-8 \pm \sqrt{64 + 60}}{-2} \][/tex]
[tex]\[ x = \frac{-8 \pm \sqrt{124}}{-2} \][/tex]
6. Simplify the square root and the expression:
[tex]\[ \sqrt{124} \text{ can be written as } \sqrt{4 \cdot 31} = 2\sqrt{31} \][/tex]
[tex]\[ x = \frac{-8 \pm 2\sqrt{31}}{-2} \][/tex]
7. Factor out the common term in the numerator:
[tex]\[ x = \frac{-8 \pm 2\sqrt{31}}{-2} = \frac{-8}{-2} \pm \frac{2\sqrt{31}}{-2} \][/tex]
Simplify both fractions:
[tex]\[ x = 4 \mp \sqrt{31} \][/tex]
Thus, the solutions are:
[tex]\[ x = 4 - \sqrt{31} \quad \text{and} \quad x = 4 + \sqrt{31} \][/tex]
Now, let's match these solutions to the given options. The correct answer is:
A. [tex]\( 4 \pm \sqrt{62} \)[/tex]
B. [tex]\( 4 \pm \sqrt{3 I} \)[/tex]
C. [tex]\( -4 \pm \sqrt{62} \)[/tex]
D. [tex]\( -1 \pm \sqrt{31} \)[/tex]
None of the options match our solutions [tex]\(4 \pm \sqrt{31}\)[/tex], which suggests there may be an error in the provided answer choices. Since we know our calculations are correct, it's possible the correct answer isn't listed.
However, based on the solution we obtained, the final values for [tex]\( x \)[/tex] are:
[tex]\[ 4 - \sqrt{31} \quad \text{and} \quad 4 + \sqrt{31} \][/tex]
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