Pre-Algebra Examples

$5 x (x + 3) = 4 (x - 4)$

Step 1

Simplify $5 x (x + 3)$ .

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Step 1.1

Rewrite.

$0 + 0 + 5 x (x + 3) = 4 (x - 4)$

Step 1.2

Simplify by adding zeros.

$5 x (x + 3) = 4 (x - 4)$

Step 1.3

Apply the distributive property.

$5 x \cdot x + 5 x \cdot 3 = 4 (x - 4)$

Step 1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 1.4.1

Move $x$ .

$5 (x \cdot x) + 5 x \cdot 3 = 4 (x - 4)$

Step 1.4.2

Multiply $x$ by $x$ .

$5 x^{2} + 5 x \cdot 3 = 4 (x - 4)$

Step 1.5

Multiply $3$ by $5$ .

$5 x^{2} + 15 x = 4 (x - 4)$

Step 2

Simplify $4 (x - 4)$ .

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Step 2.1

Apply the distributive property.

$5 x^{2} + 15 x = 4 x + 4 \cdot - 4$

Step 2.2

Multiply $4$ by $- 4$ .

$5 x^{2} + 15 x = 4 x - 16$

Step 3

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

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Step 3.1

Subtract $4 x$ from both sides of the equation.

$5 x^{2} + 15 x - 4 x = - 16$

Step 3.2

Subtract $4 x$ from $15 x$ .

$5 x^{2} + 11 x = - 16$

Step 4

Add $16$ to both sides of the equation.

$5 x^{2} + 11 x + 16 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 5$ , $b = 11$ , and $c = 16$ into the quadratic formula and solve for $x$ .

$\frac{- 11 \pm \sqrt{11^{2} - 4 \cdot (5 \cdot 16)}}{2 \cdot 5}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $11$ to the power of $2$ .

$x = \frac{- 11 \pm \sqrt{121 - 4 \cdot 5 \cdot 16}}{2 \cdot 5}$

Step 7.1.2

Multiply $- 4 \cdot 5 \cdot 16$ .

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Step 7.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 11 \pm \sqrt{121 - 20 \cdot 16}}{2 \cdot 5}$

Step 7.1.2.2

Multiply $- 20$ by $16$ .

$x = \frac{- 11 \pm \sqrt{121 - 320}}{2 \cdot 5}$

Step 7.1.3

Subtract $320$ from $121$ .

$x = \frac{- 11 \pm \sqrt{- 199}}{2 \cdot 5}$

Step 7.1.4

Rewrite $- 199$ as $- 1 (199)$ .

$x = \frac{- 11 \pm \sqrt{- 1 \cdot 199}}{2 \cdot 5}$

Step 7.1.5

Rewrite $\sqrt{- 1 (199)}$ as $\sqrt{- 1} \cdot \sqrt{199}$ .

$x = \frac{- 11 \pm \sqrt{- 1} \cdot \sqrt{199}}{2 \cdot 5}$

Step 7.1.6

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

$x = \frac{- 11 \pm i \sqrt{199}}{2 \cdot 5}$

Step 7.2

Multiply $2$ by $5$ .

$x = \frac{- 11 \pm i \sqrt{199}}{10}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{11 - i \sqrt{199}}{10}, - \frac{11 + i \sqrt{199}}{10}$