Finite Math Examples

$x (x + 3) = 5 x^{2} + 2$

Step 1

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

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify $x (x + 3)$ .

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Step 1.1.1.1

Apply the distributive property.

$x \cdot x + x \cdot 3 = 5 x^{2} + 2$

Step 1.1.1.2

Simplify the expression.

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Step 1.1.1.2.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 3 = 5 x^{2} + 2$

Step 1.1.1.2.2

Move $3$ to the left of $x$ .

$x^{2} + 3 x = 5 x^{2} + 2$

Step 1.2

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

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Step 1.2.1

Subtract $5 x^{2}$ from both sides of the equation.

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

Step 1.2.2

Subtract $2$ from both sides of the equation.

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

Step 1.3

Subtract $5 x^{2}$ from $x^{2}$ .

$- 4 x^{2} + 3 x - 2 = 0$

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 $a = - 4$ , $b = 3$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

Multiply $- 4 \cdot - 4 \cdot - 2$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 4$ .

$x = \frac{- 3 \pm \sqrt{9 + 16 \cdot - 2}}{2 \cdot - 4}$

Step 4.1.2.2

Multiply $16$ by $- 2$ .

$x = \frac{- 3 \pm \sqrt{9 - 32}}{2 \cdot - 4}$

Step 4.1.3

Subtract $32$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 23}}{2 \cdot - 4}$

Step 4.1.4

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

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 23}}{2 \cdot - 4}$

Step 4.1.5

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

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot - 4}$

Step 4.1.6

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

$x = \frac{- 3 \pm i \sqrt{23}}{2 \cdot - 4}$

Step 4.2

Multiply $2$ by $- 4$ .

$x = \frac{- 3 \pm i \sqrt{23}}{- 8}$

Step 4.3

Simplify $\frac{- 3 \pm i \sqrt{23}}{- 8}$ .

$x = \frac{3 \pm i \sqrt{23}}{8}$