Finite Math Examples

$5 x^{2} - 2 x + 13 = 18 x + 1$

Step 1

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

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Step 1.1

Subtract $18 x$ from both sides of the equation.

$5 x^{2} - 2 x + 13 - 18 x = 1$

Step 1.2

Subtract $18 x$ from $- 2 x$ .

$5 x^{2} - 20 x + 13 = 1$

Step 2

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

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Step 2.1

Subtract $1$ from both sides of the equation.

$5 x^{2} - 20 x + 13 - 1 = 0$

Step 2.2

Subtract $1$ from $13$ .

$5 x^{2} - 20 x + 12 = 0$

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 $a = 5$ , $b = - 20$ , and $c = 12$ into the quadratic formula and solve for $x$ .

$\frac{20 \pm \sqrt{{(- 20)}^{2} - 4 \cdot (5 \cdot 12)}}{2 \cdot 5}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 20$ to the power of $2$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 5 \cdot 12}}{2 \cdot 5}$

Step 5.1.2

Multiply $- 4 \cdot 5 \cdot 12$ .

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Step 5.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{20 \pm \sqrt{400 - 20 \cdot 12}}{2 \cdot 5}$

Step 5.1.2.2

Multiply $- 20$ by $12$ .

$x = \frac{20 \pm \sqrt{400 - 240}}{2 \cdot 5}$

Step 5.1.3

Subtract $240$ from $400$ .

$x = \frac{20 \pm \sqrt{160}}{2 \cdot 5}$

Step 5.1.4

Rewrite $160$ as $4^{2} \cdot 10$ .

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Step 5.1.4.1

Factor $16$ out of $160$ .

$x = \frac{20 \pm \sqrt{16 (10)}}{2 \cdot 5}$

Step 5.1.4.2

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

$x = \frac{20 \pm \sqrt{4^{2} \cdot 10}}{2 \cdot 5}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{20 \pm 4 \sqrt{10}}{2 \cdot 5}$

Step 5.2

Multiply $2$ by $5$ .

$x = \frac{20 \pm 4 \sqrt{10}}{10}$

Step 5.3

Simplify $\frac{20 \pm 4 \sqrt{10}}{10}$ .

$x = \frac{10 \pm 2 \sqrt{10}}{5}$

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 $- 20$ to the power of $2$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 5 \cdot 12}}{2 \cdot 5}$

Step 6.1.2

Multiply $- 4 \cdot 5 \cdot 12$ .

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Step 6.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{20 \pm \sqrt{400 - 20 \cdot 12}}{2 \cdot 5}$

Step 6.1.2.2

Multiply $- 20$ by $12$ .

$x = \frac{20 \pm \sqrt{400 - 240}}{2 \cdot 5}$

Step 6.1.3

Subtract $240$ from $400$ .

$x = \frac{20 \pm \sqrt{160}}{2 \cdot 5}$

Step 6.1.4

Rewrite $160$ as $4^{2} \cdot 10$ .

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Step 6.1.4.1

Factor $16$ out of $160$ .

$x = \frac{20 \pm \sqrt{16 (10)}}{2 \cdot 5}$

Step 6.1.4.2

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

$x = \frac{20 \pm \sqrt{4^{2} \cdot 10}}{2 \cdot 5}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{20 \pm 4 \sqrt{10}}{2 \cdot 5}$

Step 6.2

Multiply $2$ by $5$ .

$x = \frac{20 \pm 4 \sqrt{10}}{10}$

Step 6.3

Simplify $\frac{20 \pm 4 \sqrt{10}}{10}$ .

$x = \frac{10 \pm 2 \sqrt{10}}{5}$

Step 6.4

Change the $\pm$ to $+$ .

$x = \frac{10 + 2 \sqrt{10}}{5}$

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 $- 20$ to the power of $2$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 5 \cdot 12}}{2 \cdot 5}$

Step 7.1.2

Multiply $- 4 \cdot 5 \cdot 12$ .

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Step 7.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{20 \pm \sqrt{400 - 20 \cdot 12}}{2 \cdot 5}$

Step 7.1.2.2

Multiply $- 20$ by $12$ .

$x = \frac{20 \pm \sqrt{400 - 240}}{2 \cdot 5}$

Step 7.1.3

Subtract $240$ from $400$ .

$x = \frac{20 \pm \sqrt{160}}{2 \cdot 5}$

Step 7.1.4

Rewrite $160$ as $4^{2} \cdot 10$ .

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Step 7.1.4.1

Factor $16$ out of $160$ .

$x = \frac{20 \pm \sqrt{16 (10)}}{2 \cdot 5}$

Step 7.1.4.2

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

$x = \frac{20 \pm \sqrt{4^{2} \cdot 10}}{2 \cdot 5}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{20 \pm 4 \sqrt{10}}{2 \cdot 5}$

Step 7.2

Multiply $2$ by $5$ .

$x = \frac{20 \pm 4 \sqrt{10}}{10}$

Step 7.3

Simplify $\frac{20 \pm 4 \sqrt{10}}{10}$ .

$x = \frac{10 \pm 2 \sqrt{10}}{5}$

Step 7.4

Change the $\pm$ to $-$ .

$x = \frac{10 - 2 \sqrt{10}}{5}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{10 + 2 \sqrt{10}}{5}, \frac{10 - 2 \sqrt{10}}{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{10 + 2 \sqrt{10}}{5}, \frac{10 - 2 \sqrt{10}}{5}$

Decimal Form:

$x = 3.26491106 \dots, 0.73508893 \dots$