Pre-Algebra Examples

$(x) = - 3 x^{2} - 30 x - 74$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- 3 x^{2} - 30 x - 74 = x$

Step 2

Move all terms containing $x$ to the left side of the equation.

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Step 2.1

Subtract $x$ from both sides of the equation.

$- 3 x^{2} - 30 x - 74 - x = 0$

Step 2.2

Subtract $x$ from $- 30 x$ .

$- 3 x^{2} - 31 x - 74 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = - 3$ , $b = - 31$ , and $c = - 74$ into the quadratic formula and solve for $x$ .

$\frac{31 \pm \sqrt{{(- 31)}^{2} - 4 \cdot (- 3 \cdot - 74)}}{2 \cdot - 3}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 31$ to the power of $2$ .

$x = \frac{31 \pm \sqrt{961 - 4 \cdot - 3 \cdot - 74}}{2 \cdot - 3}$

Step 5.1.2

Multiply $- 4 \cdot - 3 \cdot - 74$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{31 \pm \sqrt{961 + 12 \cdot - 74}}{2 \cdot - 3}$

Step 5.1.2.2

Multiply $12$ by $- 74$ .

$x = \frac{31 \pm \sqrt{961 - 888}}{2 \cdot - 3}$

Step 5.1.3

Subtract $888$ from $961$ .

$x = \frac{31 \pm \sqrt{73}}{2 \cdot - 3}$

Step 5.2

Multiply $2$ by $- 3$ .

$x = \frac{31 \pm \sqrt{73}}{- 6}$

Step 5.3

Move the negative in front of the fraction.

$x = - \frac{31 \pm \sqrt{73}}{6}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 31$ to the power of $2$ .

$x = \frac{31 \pm \sqrt{961 - 4 \cdot - 3 \cdot - 74}}{2 \cdot - 3}$

Step 6.1.2

Multiply $- 4 \cdot - 3 \cdot - 74$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{31 \pm \sqrt{961 + 12 \cdot - 74}}{2 \cdot - 3}$

Step 6.1.2.2

Multiply $12$ by $- 74$ .

$x = \frac{31 \pm \sqrt{961 - 888}}{2 \cdot - 3}$

Step 6.1.3

Subtract $888$ from $961$ .

$x = \frac{31 \pm \sqrt{73}}{2 \cdot - 3}$

Step 6.2

Multiply $2$ by $- 3$ .

$x = \frac{31 \pm \sqrt{73}}{- 6}$

Step 6.3

Move the negative in front of the fraction.

$x = - \frac{31 \pm \sqrt{73}}{6}$

Step 6.4

Change the $\pm$ to $+$ .

$x = - \frac{31 + \sqrt{73}}{6}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 31$ to the power of $2$ .

$x = \frac{31 \pm \sqrt{961 - 4 \cdot - 3 \cdot - 74}}{2 \cdot - 3}$

Step 7.1.2

Multiply $- 4 \cdot - 3 \cdot - 74$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{31 \pm \sqrt{961 + 12 \cdot - 74}}{2 \cdot - 3}$

Step 7.1.2.2

Multiply $12$ by $- 74$ .

$x = \frac{31 \pm \sqrt{961 - 888}}{2 \cdot - 3}$

Step 7.1.3

Subtract $888$ from $961$ .

$x = \frac{31 \pm \sqrt{73}}{2 \cdot - 3}$

Step 7.2

Multiply $2$ by $- 3$ .

$x = \frac{31 \pm \sqrt{73}}{- 6}$

Step 7.3

Move the negative in front of the fraction.

$x = - \frac{31 \pm \sqrt{73}}{6}$

Step 7.4

Change the $\pm$ to $-$ .

$x = - \frac{31 - \sqrt{73}}{6}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{31 + \sqrt{73}}{6}, - \frac{31 - \sqrt{73}}{6}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{31 + \sqrt{73}}{6}, - \frac{31 - \sqrt{73}}{6}$

Decimal Form:

$x = - 6.59066729 \dots, - 3.74266604 \dots$