Question

Complex Numbers & Higher Order Polynomials Quick Check
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Question
Which answer shows the conjugate pair solutions to the equation x2−3x+8=−2
?(1 point)
Responses

−32±i31√2
negative Start Fraction 3 over 2 End Fraction plus or minus Start Fraction i Start Root 31 End Root over 2 End Fraction

32±i31√2
Start Fraction 3 over 2 End Fraction plus or minus Start Fraction i Start Root 31 End Root over 2 End Fraction

32±i15√2
Start Fraction 3 over 2 End Fraction plus or minus Start Fraction i Start Root 15 End Root over 2 End Fraction

32±i23√2

Answer 1

To solve the equation \(x^2 - 3x + 8 = -2\), first we can rewrite it in standard form by adding 2 to both sides:

\[
x^2 - 3x + 10 = 0
\]

Next, we can use the quadratic formula to find the roots, which is given by:

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

In this case, \(a = 1\), \(b = -3\), and \(c = 10\). Plugging in these values:

1. Calculate \(b^2 - 4ac\):

\[
(-3)^2 - 4(1)(10) = 9 - 40 = -31
\]

Since the discriminant is negative, the roots will be complex. Now substituting back into the quadratic formula:

\[
x = \frac{-(-3) \pm \sqrt{-31}}{2(1)} = \frac{3 \pm i\sqrt{31}}{2}
\]

Thus, the conjugate pair solutions are:

\[
x = \frac{3}{2} \pm \frac{i\sqrt{31}}{2}
\]

Looking at the provided options, the correct answer is:

\[
\frac{3}{2} \pm \frac{i\sqrt{31}}{2}
\]

So the correct choice is:

**negative Start Fraction 3 over 2 End Fraction plus or minus Start Fraction i Start Root 31 End Root over 2 End Fraction**