Algebra Examples

$- x^{2} + 3 x - 8 = 0$

Step 1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2

Substitute the values $a = - 1$ , $b = 3$ , and $c = - 8$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (- 1 \cdot - 8)}}{2 \cdot - 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 1 \cdot - 8}}{2 \cdot - 1}$

Step 3.1.2

Multiply $- 4 \cdot - 1 \cdot - 8$ .

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Step 3.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{9 + 4 \cdot - 8}}{2 \cdot - 1}$

Step 3.1.2.2

Multiply $4$ by $- 8$ .

$x = \frac{- 3 \pm \sqrt{9 - 32}}{2 \cdot - 1}$

Step 3.1.3

Subtract $32$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 23}}{2 \cdot - 1}$

Step 3.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 23}}{2 \cdot - 1}$

Step 3.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot - 1}$

Step 3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 3 \pm i \sqrt{23}}{2 \cdot - 1}$

Step 3.2

Multiply $2$ by $- 1$ .

$x = \frac{- 3 \pm i \sqrt{23}}{- 2}$

Step 3.3

Simplify $\frac{- 3 \pm i \sqrt{23}}{- 2}$ .

$x = \frac{3 \pm i \sqrt{23}}{2}$

Step 4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 1 \cdot - 8}}{2 \cdot - 1}$

Step 4.1.2

Multiply $- 4 \cdot - 1 \cdot - 8$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{9 + 4 \cdot - 8}}{2 \cdot - 1}$

Step 4.1.2.2

Multiply $4$ by $- 8$ .

$x = \frac{- 3 \pm \sqrt{9 - 32}}{2 \cdot - 1}$

Step 4.1.3

Subtract $32$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 23}}{2 \cdot - 1}$

Step 4.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 23}}{2 \cdot - 1}$

Step 4.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot - 1}$

Step 4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 3 \pm i \sqrt{23}}{2 \cdot - 1}$

Step 4.2

Multiply $2$ by $- 1$ .

$x = \frac{- 3 \pm i \sqrt{23}}{- 2}$

Step 4.3

Simplify $\frac{- 3 \pm i \sqrt{23}}{- 2}$ .

$x = \frac{3 \pm i \sqrt{23}}{2}$

Step 4.4

Change the $\pm$ to $+$ .

$x = \frac{3 + i \sqrt{23}}{2}$

Step 5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 1 \cdot - 8}}{2 \cdot - 1}$

Step 5.1.2

Multiply $- 4 \cdot - 1 \cdot - 8$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 3 \pm \sqrt{9 + 4 \cdot - 8}}{2 \cdot - 1}$

Step 5.1.2.2

Multiply $4$ by $- 8$ .

$x = \frac{- 3 \pm \sqrt{9 - 32}}{2 \cdot - 1}$

Step 5.1.3

Subtract $32$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 23}}{2 \cdot - 1}$

Step 5.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 23}}{2 \cdot - 1}$

Step 5.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot - 1}$

Step 5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 3 \pm i \sqrt{23}}{2 \cdot - 1}$

Step 5.2

Multiply $2$ by $- 1$ .

$x = \frac{- 3 \pm i \sqrt{23}}{- 2}$

Step 5.3

Simplify $\frac{- 3 \pm i \sqrt{23}}{- 2}$ .

$x = \frac{3 \pm i \sqrt{23}}{2}$

Step 5.4

Change the $\pm$ to $-$ .

$x = \frac{3 - i \sqrt{23}}{2}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{3 + i \sqrt{23}}{2}, \frac{3 - i \sqrt{23}}{2}$

Step 7

The domain is the set of all valid $x$ values.

${x | x = \frac{3 + i \sqrt{23}}{2}, \frac{3 - i \sqrt{23}}{2}}$

Step 8

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 9

Determine the domain and range.

Domain: ${x | x = \frac{3 + i \sqrt{23}}{2}, \frac{3 - i \sqrt{23}}{2}}$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 10