Finite Math Examples

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

Step 1

Simplify the equation into a proper form to complete the square.

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Step 1.1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1.1

Subtract $13$ from both sides of the equation.

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

Step 1.1.2

Subtract $13$ from $1$ .

$5 x^{2} - 2 x = 18 x - 12$

Step 1.2

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

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Step 1.2.1

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

$5 x^{2} - 2 x - 18 x = - 12$

Step 1.2.2

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

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

Step 2

Divide each term in $5 x^{2} - 20 x = - 12$ by $5$ and simplify.

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Step 2.1

Divide each term in $5 x^{2} - 20 x = - 12$ by $5$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $5$ .

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Step 2.2.1.1.1

Cancel the common factor.

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

Step 2.2.1.1.2

Divide $x^{2}$ by $1$ .

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

Step 2.2.1.2

Cancel the common factor of $- 20$ and $5$ .

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Step 2.2.1.2.1

Factor $5$ out of $- 20 x$ .

$x^{2} + \frac{5 (- 4 x)}{5} = \frac{- 12}{5}$

Step 2.2.1.2.2

Cancel the common factors.

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Step 2.2.1.2.2.1

Factor $5$ out of $5$ .

$x^{2} + \frac{5 (- 4 x)}{5 (1)} = \frac{- 12}{5}$

Step 2.2.1.2.2.2

Cancel the common factor.

$x^{2} + \frac{5 (- 4 x)}{5 \cdot 1} = \frac{- 12}{5}$

Step 2.2.1.2.2.3

Rewrite the expression.

$x^{2} + \frac{- 4 x}{1} = \frac{- 12}{5}$

Step 2.2.1.2.2.4

Divide $- 4 x$ by $1$ .

$x^{2} - 4 x = \frac{- 12}{5}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$x^{2} - 4 x = - \frac{12}{5}$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(- 2)}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} - 4 x + {(- 2)}^{2} = - \frac{12}{5} + {(- 2)}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

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

$x^{2} - 4 x + 4 = - \frac{12}{5} + {(- 2)}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- \frac{12}{5} + {(- 2)}^{2}$ .

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Step 5.2.1.1

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

$x^{2} - 4 x + 4 = - \frac{12}{5} + 4$

Step 5.2.1.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x^{2} - 4 x + 4 = - \frac{12}{5} + 4 \cdot \frac{5}{5}$

Step 5.2.1.3

Combine $4$ and $\frac{5}{5}$ .

$x^{2} - 4 x + 4 = - \frac{12}{5} + \frac{4 \cdot 5}{5}$

Step 5.2.1.4

Combine the numerators over the common denominator.

$x^{2} - 4 x + 4 = \frac{- 12 + 4 \cdot 5}{5}$

Step 5.2.1.5

Simplify the numerator.

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Step 5.2.1.5.1

Multiply $4$ by $5$ .

$x^{2} - 4 x + 4 = \frac{- 12 + 20}{5}$

Step 5.2.1.5.2

Add $- 12$ and $20$ .

$x^{2} - 4 x + 4 = \frac{8}{5}$

Step 6

Factor the perfect trinomial square into ${(x - 2)}^{2}$ .

${(x - 2)}^{2} = \frac{8}{5}$

Step 7

Solve the equation for $x$ .

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Step 7.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x - 2 = \pm \sqrt{\frac{8}{5}}$

Step 7.2

Simplify $\pm \sqrt{\frac{8}{5}}$ .

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Step 7.2.1

Rewrite $\sqrt{\frac{8}{5}}$ as $\frac{\sqrt{8}}{\sqrt{5}}$ .

$x - 2 = \pm \frac{\sqrt{8}}{\sqrt{5}}$

Step 7.2.2

Simplify the numerator.

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Step 7.2.2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 7.2.2.1.1

Factor $4$ out of $8$ .

$x - 2 = \pm \frac{\sqrt{4 (2)}}{\sqrt{5}}$

Step 7.2.2.1.2

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

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

Step 7.2.2.2

Pull terms out from under the radical.

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

Step 7.2.3

Multiply $\frac{2 \sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 7.2.4

Combine and simplify the denominator.

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Step 7.2.4.1

Multiply $\frac{2 \sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 7.2.4.2

Raise $\sqrt{5}$ to the power of $1$ .

$x - 2 = \pm \frac{2 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 7.2.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$x - 2 = \pm \frac{2 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 7.2.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x - 2 = \pm \frac{2 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 7.2.4.5

Add $1$ and $1$ .

$x - 2 = \pm \frac{2 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 7.2.4.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 7.2.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x - 2 = \pm \frac{2 \sqrt{2} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 7.2.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 7.2.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$x - 2 = \pm \frac{2 \sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 7.2.4.6.4

Cancel the common factor of $2$ .

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Step 7.2.4.6.4.1

Cancel the common factor.

$x - 2 = \pm \frac{2 \sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 7.2.4.6.4.2

Rewrite the expression.

$x - 2 = \pm \frac{2 \sqrt{2} \sqrt{5}}{5^{1}}$

Step 7.2.4.6.5

Evaluate the exponent.

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

Step 7.2.5

Simplify the numerator.

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Step 7.2.5.1

Combine using the product rule for radicals.

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

Step 7.2.5.2

Multiply $5$ by $2$ .

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

Step 7.3

Add $2$ to both sides of the equation.

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 3.26491106 \dots, 0.73508893 \dots$