QuestionSolve the equation $x^{2}+3 x+8=0$ and express solutions as $a+b i$ .

Studdy Solution

STEP 1

Assumptions1. The equation is a quadratic equation of the form $ax^ + bx + c =0$ . The coefficients are a=1, b=3, and c=83. We are asked to find the solutions in the form $a+bi$ , where $a$ and $b$ are real numbers and $i$ is the imaginary unit

STEP 2

The quadratic formula is used to find the solutions of a quadratic equation. The formula is given by $x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$

STEP 3

Now, plug in the given values for the coefficients a, b, and c into the quadratic formula.
$x = \frac{-3 \pm \sqrt{3^2 -18}}{2*1}$

STEP 4

implify the equation.
$x = \frac{-3 \pm \sqrt{9 -32}}{2}$

STEP 5

Calculate the value under the square root.
$x = \frac{-3 \pm \sqrt{-23}}{2}$

STEP 6

Since the value under the square root is negative, we have an imaginary solution. We can express the square root of -23 as $i\sqrt{23}$ , where $i$ is the imaginary unit.
$x = \frac{-3 \pm i\sqrt{23}}{2}$

STEP 7

Finally, we can write the solutions in the form $a+bi$ .
$x = -\frac{3}{2} \pm \frac{1}{2}i\sqrt{23}$ So the solutions of the equation are $x = -\frac{3}{2} + \frac{1}{2}i\sqrt{23}$ and $x = -\frac{3}{2} - \frac{1}{2}i\sqrt{23}$ .

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