Algebra Examples

$3 x^{2} + 30 x = - 87$

Step 1

Add $87$ to both sides of the equation.

$3 x^{2} + 30 x + 87 = 0$

Step 2

Factor $3$ out of $3 x^{2} + 30 x + 87$ .

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Step 2.1

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) + 30 x + 87 = 0$

Step 2.2

Factor $3$ out of $30 x$ .

$3 (x^{2}) + 3 (10 x) + 87 = 0$

Step 2.3

Factor $3$ out of $87$ .

$3 x^{2} + 3 (10 x) + 3 \cdot 29 = 0$

Step 2.4

Factor $3$ out of $3 x^{2} + 3 (10 x)$ .

$3 (x^{2} + 10 x) + 3 \cdot 29 = 0$

Step 2.5

Factor $3$ out of $3 (x^{2} + 10 x) + 3 \cdot 29$ .

$3 (x^{2} + 10 x + 29) = 0$

Step 3

Divide each term in $3 (x^{2} + 10 x + 29) = 0$ by $3$ and simplify.

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Step 3.1

Divide each term in $3 (x^{2} + 10 x + 29) = 0$ by $3$ .

$\frac{3 (x^{2} + 10 x + 29)}{3} = \frac{0}{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3 (x^{2} + 10 x + 29)}{3} = \frac{0}{3}$

Step 3.2.1.2

Divide $x^{2} + 10 x + 29$ by $1$ .

$x^{2} + 10 x + 29 = \frac{0}{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $3$ .

$x^{2} + 10 x + 29 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 1$ , $b = 10$ , and $c = 29$ into the quadratic formula and solve for $x$ .

$\frac{- 10 \pm \sqrt{10^{2} - 4 \cdot (1 \cdot 29)}}{2 \cdot 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $10$ to the power of $2$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 1 \cdot 29}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot 29$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 29}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $29$ .

$x = \frac{- 10 \pm \sqrt{100 - 116}}{2 \cdot 1}$

Step 6.1.3

Subtract $116$ from $100$ .

$x = \frac{- 10 \pm \sqrt{- 16}}{2 \cdot 1}$

Step 6.1.4

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{- 10 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Step 6.1.5

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{- 10 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

Step 6.1.6

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

$x = \frac{- 10 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

Step 6.1.7

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 10 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

Step 6.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 10 \pm i \cdot 4}{2 \cdot 1}$

Step 6.1.9

Move $4$ to the left of $i$ .

$x = \frac{- 10 \pm 4 i}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{- 10 \pm 4 i}{2}$

Step 6.3

Simplify $\frac{- 10 \pm 4 i}{2}$ .

$x = - 5 \pm 2 i$

Step 7