Which answer shows the conjugate pair solutions to the equation x2−3x+8=−2

?(1 point)
Responses

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
Start Fraction 3 over 2 End Fraction plus or minus Start Fraction i Start Root 23 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±i31√2

1 answer

To solve the equation \( x^2 - 3x + 8 = -2 \), we first move \(-2\) to the left side, resulting in:

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

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

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

Here, \( a = 1 \), \( b = -3 \), and \( c = 10 \). We can substitute these values into the formula:

  1. Calculate \( b^2 - 4ac \): \[ (-3)^2 - 4(1)(10) = 9 - 40 = -31 \]

  2. Now substitute the values into the quadratic formula: \[ x = \frac{-(-3) \pm \sqrt{-31}}{2 \cdot 1} = \frac{3 \pm \sqrt{-31}}{2} \]

  3. Since \( \sqrt{-31} = i\sqrt{31} \), we can rewrite the solutions as: \[ x = \frac{3 \pm i\sqrt{31}}{2} \]

This means the solutions can be expressed as: \[ x = \frac{3}{2} \pm \frac{i\sqrt{31}}{2} \]

So, the correct answer showing the conjugate pair solutions is:

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