Algebra Examples

$- 8 x + 30 = - 3 x^{2} + 5 x$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Move all the expressions to the left side of the equation.

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Step 1.1.1

Add $3 x^{2}$ to both sides of the equation.

$- 8 x + 30 + 3 x^{2} = 5 x$

Step 1.1.2

Subtract $5 x$ from both sides of the equation.

$- 8 x + 30 + 3 x^{2} - 5 x = 0$

Step 1.2

Subtract $5 x$ from $- 8 x$ .

$- 13 x + 30 + 3 x^{2} = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 3$ , $b = - 13$ , and $c = 30$ into the quadratic formula and solve for $x$ .

$\frac{13 \pm \sqrt{{(- 13)}^{2} - 4 \cdot (3 \cdot 30)}}{2 \cdot 3}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 13$ to the power of $2$ .

$x = \frac{13 \pm \sqrt{169 - 4 \cdot 3 \cdot 30}}{2 \cdot 3}$

Step 4.1.2

Multiply $- 4 \cdot 3 \cdot 30$ .

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Step 4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{13 \pm \sqrt{169 - 12 \cdot 30}}{2 \cdot 3}$

Step 4.1.2.2

Multiply $- 12$ by $30$ .

$x = \frac{13 \pm \sqrt{169 - 360}}{2 \cdot 3}$

Step 4.1.3

Subtract $360$ from $169$ .

$x = \frac{13 \pm \sqrt{- 191}}{2 \cdot 3}$

Step 4.1.4

Rewrite $- 191$ as $- 1 (191)$ .

$x = \frac{13 \pm \sqrt{- 1 \cdot 191}}{2 \cdot 3}$

Step 4.1.5

Rewrite $\sqrt{- 1 (191)}$ as $\sqrt{- 1} \cdot \sqrt{191}$ .

$x = \frac{13 \pm \sqrt{- 1} \cdot \sqrt{191}}{2 \cdot 3}$

Step 4.1.6

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

$x = \frac{13 \pm i \sqrt{191}}{2 \cdot 3}$

Step 4.2

Multiply $2$ by $3$ .

$x = \frac{13 \pm i \sqrt{191}}{6}$