Which answer shows the conjugate pair solutions to the equation x2−3x+8=−2 ?(1 point)

Question

Responses

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±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±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

Answer 1

To find the conjugate pair solutions to the equation $x^2-3x+8=-2$, we first rewrite the equation in standard form:

$$x^2-3x+8+2=0⟹x^2-3x+10=0.$$

Next, we can apply the quadratic formula, which is given by:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a},$$

where $a=1$, $b=-3$, and $c=10$.

Calculating the discriminant:

$$b^2-4ac=(-3{)}^2-4(1)(10)=9-40=-31.$$

Since the discriminant is negative, we can conclude that there are two complex conjugate solutions. We now can substitute back into the quadratic formula:

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

Thus, the solutions can be written as:

$$\frac{3}{2}\pm \frac{i\sqrt{31}}{2}.$$

So the correct answer showing the conjugate pair solutions for the equation $x^2-3x+8=-2$ from the provided options is:

Negative $\frac{3}{2}$ plus or minus $\frac{i\sqrt{31}}{2}$.