Finite Math Examples

$\sqrt{\frac{x - 8 + 4}{2}} + \sqrt{36} - 4 x = 3^{2} (\frac{1}{6})$

Step 1

Solve for $\sqrt{\frac{x - 8 + 4}{2}}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite $36$ as $6^{2}$ .

$\sqrt{\frac{x - 8 + 4}{2}} + \sqrt{6^{2}} - 4 x = 3^{2} (\frac{1}{6})$

Step 1.1.2

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

$\sqrt{\frac{x - 8 + 4}{2}} + 6 - 4 x = 3^{2} (\frac{1}{6})$

Step 1.2

Simplify $3^{2} (\frac{1}{6})$ .

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Step 1.2.1

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

$\sqrt{\frac{x - 8 + 4}{2}} + 6 - 4 x = 9 (\frac{1}{6})$

Step 1.2.2

Cancel the common factor of $3$ .

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Step 1.2.2.1

Factor $3$ out of $9$ .

$\sqrt{\frac{x - 8 + 4}{2}} + 6 - 4 x = 3 (3) \frac{1}{6}$

Step 1.2.2.2

Factor $3$ out of $6$ .

$\sqrt{\frac{x - 8 + 4}{2}} + 6 - 4 x = 3 \cdot 3 \frac{1}{3 \cdot 2}$

Step 1.2.2.3

Cancel the common factor.

$\sqrt{\frac{x - 8 + 4}{2}} + 6 - 4 x = 3 \cdot 3 \frac{1}{3 \cdot 2}$

Step 1.2.2.4

Rewrite the expression.

$\sqrt{\frac{x - 8 + 4}{2}} + 6 - 4 x = 3 (\frac{1}{2})$

Step 1.2.3

Combine $3$ and $\frac{1}{2}$ .

$\sqrt{\frac{x - 8 + 4}{2}} + 6 - 4 x = \frac{3}{2}$

Step 1.3

Move all terms not containing $\sqrt{\frac{x - 8 + 4}{2}}$ to the right side of the equation.

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Step 1.3.1

Subtract $6$ from both sides of the equation.

$\sqrt{\frac{x - 8 + 4}{2}} - 4 x = \frac{3}{2} - 6$

Step 1.3.2

Add $4 x$ to both sides of the equation.

$\sqrt{\frac{x - 8 + 4}{2}} = \frac{3}{2} - 6 + 4 x$

Step 1.3.3

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{x - 8 + 4}{2}} = \frac{3}{2} - 6 \cdot \frac{2}{2} + 4 x$

Step 1.3.4

Combine $- 6$ and $\frac{2}{2}$ .

$\sqrt{\frac{x - 8 + 4}{2}} = \frac{3}{2} + \frac{- 6 \cdot 2}{2} + 4 x$

Step 1.3.5

Combine the numerators over the common denominator.

$\sqrt{\frac{x - 8 + 4}{2}} = \frac{3 - 6 \cdot 2}{2} + 4 x$

Step 1.3.6

Simplify the numerator.

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Step 1.3.6.1

Multiply $- 6$ by $2$ .

$\sqrt{\frac{x - 8 + 4}{2}} = \frac{3 - 12}{2} + 4 x$

Step 1.3.6.2

Subtract $12$ from $3$ .

$\sqrt{\frac{x - 8 + 4}{2}} = \frac{- 9}{2} + 4 x$

Step 1.3.7

Move the negative in front of the fraction.

$\sqrt{\frac{x - 8 + 4}{2}} = - \frac{9}{2} + 4 x$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{\frac{x - 8 + 4}{2}}}^{2} = {(- \frac{9}{2} + 4 x)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{x - 8 + 4}{2}}$ as ${(\frac{x - 8 + 4}{2})}^{\frac{1}{2}}$ .

${({(\frac{x - 8 + 4}{2})}^{\frac{1}{2}})}^{2} = {(- \frac{9}{2} + 4 x)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(\frac{x - 8 + 4}{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(\frac{x - 8 + 4}{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(\frac{x - 8 + 4}{2})}^{\frac{1}{2} \cdot 2} = {(- \frac{9}{2} + 4 x)}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(\frac{x - 8 + 4}{2})}^{\frac{1}{2} \cdot 2} = {(- \frac{9}{2} + 4 x)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(\frac{x - 8 + 4}{2})}^{1} = {(- \frac{9}{2} + 4 x)}^{2}$

Step 3.2.1.2

Add $- 8$ and $4$ .

${(\frac{x - 4}{2})}^{1} = {(- \frac{9}{2} + 4 x)}^{2}$

Step 3.2.1.3

Simplify.

$\frac{x - 4}{2} = {(- \frac{9}{2} + 4 x)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(- \frac{9}{2} + 4 x)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(- \frac{9}{2} + 4 x)}^{2}$ as $(- \frac{9}{2} + 4 x) (- \frac{9}{2} + 4 x)$ .

$\frac{x - 4}{2} = (- \frac{9}{2} + 4 x) (- \frac{9}{2} + 4 x)$

Step 3.3.1.2

Expand $(- \frac{9}{2} + 4 x) (- \frac{9}{2} + 4 x)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$\frac{x - 4}{2} = - \frac{9}{2} (- \frac{9}{2} + 4 x) + 4 x (- \frac{9}{2} + 4 x)$

Step 3.3.1.2.2

Apply the distributive property.

$\frac{x - 4}{2} = - \frac{9}{2} (- \frac{9}{2}) - \frac{9}{2} (4 x) + 4 x (- \frac{9}{2} + 4 x)$

Step 3.3.1.2.3

Apply the distributive property.

$\frac{x - 4}{2} = - \frac{9}{2} (- \frac{9}{2}) - \frac{9}{2} (4 x) + 4 x (- \frac{9}{2}) + 4 x (4 x)$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $- \frac{9}{2} (- \frac{9}{2})$ .

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Step 3.3.1.3.1.1.1

Multiply $- 1$ by $- 1$ .

$\frac{x - 4}{2} = 1 (\frac{9}{2}) \frac{9}{2} - \frac{9}{2} (4 x) + 4 x (- \frac{9}{2}) + 4 x (4 x)$

Step 3.3.1.3.1.1.2

Multiply $\frac{9}{2}$ by $1$ .

$\frac{x - 4}{2} = \frac{9}{2} \cdot \frac{9}{2} - \frac{9}{2} (4 x) + 4 x (- \frac{9}{2}) + 4 x (4 x)$

Step 3.3.1.3.1.1.3

Multiply $\frac{9}{2}$ by $\frac{9}{2}$ .

$\frac{x - 4}{2} = \frac{9 \cdot 9}{2 \cdot 2} - \frac{9}{2} (4 x) + 4 x (- \frac{9}{2}) + 4 x (4 x)$

Step 3.3.1.3.1.1.4

Multiply $9$ by $9$ .

$\frac{x - 4}{2} = \frac{81}{2 \cdot 2} - \frac{9}{2} (4 x) + 4 x (- \frac{9}{2}) + 4 x (4 x)$

Step 3.3.1.3.1.1.5

Multiply $2$ by $2$ .

$\frac{x - 4}{2} = \frac{81}{4} - \frac{9}{2} (4 x) + 4 x (- \frac{9}{2}) + 4 x (4 x)$

Step 3.3.1.3.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.2.1

Move the leading negative in $- \frac{9}{2}$ into the numerator.

$\frac{x - 4}{2} = \frac{81}{4} + \frac{- 9}{2} (4 x) + 4 x (- \frac{9}{2}) + 4 x (4 x)$

Step 3.3.1.3.1.2.2

Factor $2$ out of $4 x$ .

$\frac{x - 4}{2} = \frac{81}{4} + \frac{- 9}{2} (2 (2 x)) + 4 x (- \frac{9}{2}) + 4 x (4 x)$

Step 3.3.1.3.1.2.3

Cancel the common factor.

$\frac{x - 4}{2} = \frac{81}{4} + \frac{- 9}{2} (2 (2 x)) + 4 x (- \frac{9}{2}) + 4 x (4 x)$

Step 3.3.1.3.1.2.4

Rewrite the expression.

$\frac{x - 4}{2} = \frac{81}{4} - 9 (2 x) + 4 x (- \frac{9}{2}) + 4 x (4 x)$

Step 3.3.1.3.1.3

Multiply $2$ by $- 9$ .

$\frac{x - 4}{2} = \frac{81}{4} - 18 x + 4 x (- \frac{9}{2}) + 4 x (4 x)$

Step 3.3.1.3.1.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.4.1

Move the leading negative in $- \frac{9}{2}$ into the numerator.

$\frac{x - 4}{2} = \frac{81}{4} - 18 x + 4 x \frac{- 9}{2} + 4 x (4 x)$

Step 3.3.1.3.1.4.2

Factor $2$ out of $4 x$ .

$\frac{x - 4}{2} = \frac{81}{4} - 18 x + 2 (2 x) \frac{- 9}{2} + 4 x (4 x)$

Step 3.3.1.3.1.4.3

Cancel the common factor.

$\frac{x - 4}{2} = \frac{81}{4} - 18 x + 2 (2 x) \frac{- 9}{2} + 4 x (4 x)$

Step 3.3.1.3.1.4.4

Rewrite the expression.

$\frac{x - 4}{2} = \frac{81}{4} - 18 x + 2 x \cdot - 9 + 4 x (4 x)$

Step 3.3.1.3.1.5

Multiply $- 9$ by $2$ .

$\frac{x - 4}{2} = \frac{81}{4} - 18 x - 18 x + 4 x (4 x)$

Step 3.3.1.3.1.6

Rewrite using the commutative property of multiplication.

$\frac{x - 4}{2} = \frac{81}{4} - 18 x - 18 x + 4 \cdot 4 x \cdot x$

Step 3.3.1.3.1.7

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

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Step 3.3.1.3.1.7.1

Move $x$ .

$\frac{x - 4}{2} = \frac{81}{4} - 18 x - 18 x + 4 \cdot 4 (x \cdot x)$

Step 3.3.1.3.1.7.2

Multiply $x$ by $x$ .

$\frac{x - 4}{2} = \frac{81}{4} - 18 x - 18 x + 4 \cdot 4 x^{2}$

Step 3.3.1.3.1.8

Multiply $4$ by $4$ .

$\frac{x - 4}{2} = \frac{81}{4} - 18 x - 18 x + 16 x^{2}$

Step 3.3.1.3.2

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

$\frac{x - 4}{2} = \frac{81}{4} - 36 x + 16 x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Multiply both sides by $2$ .

$\frac{x - 4}{2} \cdot 2 = (\frac{81}{4} - 36 x + 16 x^{2}) \cdot 2$

Step 4.2

Simplify.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Cancel the common factor of $2$ .

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Step 4.2.1.1.1

Cancel the common factor.

$\frac{x - 4}{2} \cdot 2 = (\frac{81}{4} - 36 x + 16 x^{2}) \cdot 2$

Step 4.2.1.1.2

Rewrite the expression.

$x - 4 = (\frac{81}{4} - 36 x + 16 x^{2}) \cdot 2$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $(\frac{81}{4} - 36 x + 16 x^{2}) \cdot 2$ .

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Step 4.2.2.1.1

Apply the distributive property.

$x - 4 = \frac{81}{4} \cdot 2 - 36 x \cdot 2 + 16 x^{2} \cdot 2$

Step 4.2.2.1.2

Simplify.

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Step 4.2.2.1.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.2.1.1

Factor $2$ out of $4$ .

$x - 4 = \frac{81}{2 (2)} \cdot 2 - 36 x \cdot 2 + 16 x^{2} \cdot 2$

Step 4.2.2.1.2.1.2

Cancel the common factor.

$x - 4 = \frac{81}{2 \cdot 2} \cdot 2 - 36 x \cdot 2 + 16 x^{2} \cdot 2$

Step 4.2.2.1.2.1.3

Rewrite the expression.

$x - 4 = \frac{81}{2} - 36 x \cdot 2 + 16 x^{2} \cdot 2$

Step 4.2.2.1.2.2

Multiply $2$ by $- 36$ .

$x - 4 = \frac{81}{2} - 72 x + 16 x^{2} \cdot 2$

Step 4.2.2.1.2.3

Multiply $2$ by $16$ .

$x - 4 = \frac{81}{2} - 72 x + 32 x^{2}$

Step 4.2.2.1.3

Move $\frac{81}{2}$ .

$x - 4 = - 72 x + 32 x^{2} + \frac{81}{2}$

Step 4.2.2.1.4

Reorder $- 72 x$ and $32 x^{2}$ .

$x - 4 = 32 x^{2} - 72 x + \frac{81}{2}$

Step 4.3

Solve for $x$ .

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Step 4.3.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$32 x^{2} - 72 x + \frac{81}{2} = x - 4$

Step 4.3.2

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

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Step 4.3.2.1

Subtract $x$ from both sides of the equation.

$32 x^{2} - 72 x + \frac{81}{2} - x = - 4$

Step 4.3.2.2

Subtract $x$ from $- 72 x$ .

$32 x^{2} - 73 x + \frac{81}{2} = - 4$

Step 4.3.3

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

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Step 4.3.3.1

Add $4$ to both sides of the equation.

$32 x^{2} - 73 x + \frac{81}{2} + 4 = 0$

Step 4.3.3.2

Simplify $32 x^{2} - 73 x + \frac{81}{2} + 4$ .

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Step 4.3.3.2.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$32 x^{2} - 73 x + \frac{81}{2} + 4 \cdot \frac{2}{2} = 0$

Step 4.3.3.2.2

Combine $4$ and $\frac{2}{2}$ .

$32 x^{2} - 73 x + \frac{81}{2} + \frac{4 \cdot 2}{2} = 0$

Step 4.3.3.2.3

Combine the numerators over the common denominator.

$32 x^{2} - 73 x + \frac{81 + 4 \cdot 2}{2} = 0$

Step 4.3.3.2.4

Simplify the numerator.

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Step 4.3.3.2.4.1

Multiply $4$ by $2$ .

$32 x^{2} - 73 x + \frac{81 + 8}{2} = 0$

Step 4.3.3.2.4.2

Add $81$ and $8$ .

$32 x^{2} - 73 x + \frac{89}{2} = 0$

Step 4.3.4

Multiply through by the least common denominator $2$ , then simplify.

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Step 4.3.4.1

Apply the distributive property.

$2 (32 x^{2}) + 2 (- 73 x) + 2 (\frac{89}{2}) = 0$

Step 4.3.4.2

Simplify.

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Step 4.3.4.2.1

Multiply $32$ by $2$ .

$64 x^{2} + 2 (- 73 x) + 2 (\frac{89}{2}) = 0$

Step 4.3.4.2.2

Multiply $- 73$ by $2$ .

$64 x^{2} - 146 x + 2 (\frac{89}{2}) = 0$

Step 4.3.4.2.3

Cancel the common factor of $2$ .

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Step 4.3.4.2.3.1

Cancel the common factor.

$64 x^{2} - 146 x + 2 (\frac{89}{2}) = 0$

Step 4.3.4.2.3.2

Rewrite the expression.

$64 x^{2} - 146 x + 89 = 0$

Step 4.3.5

Use the quadratic formula to find the solutions.

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

Step 4.3.6

Substitute the values $a = 64$ , $b = - 146$ , and $c = 89$ into the quadratic formula and solve for $x$ .

$\frac{146 \pm \sqrt{{(- 146)}^{2} - 4 \cdot (64 \cdot 89)}}{2 \cdot 64}$

Step 4.3.7

Simplify.

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Step 4.3.7.1

Simplify the numerator.

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Step 4.3.7.1.1

Raise $- 146$ to the power of $2$ .

$x = \frac{146 \pm \sqrt{21316 - 4 \cdot 64 \cdot 89}}{2 \cdot 64}$

Step 4.3.7.1.2

Multiply $- 4 \cdot 64 \cdot 89$ .

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Step 4.3.7.1.2.1

Multiply $- 4$ by $64$ .

$x = \frac{146 \pm \sqrt{21316 - 256 \cdot 89}}{2 \cdot 64}$

Step 4.3.7.1.2.2

Multiply $- 256$ by $89$ .

$x = \frac{146 \pm \sqrt{21316 - 22784}}{2 \cdot 64}$

Step 4.3.7.1.3

Subtract $22784$ from $21316$ .

$x = \frac{146 \pm \sqrt{- 1468}}{2 \cdot 64}$

Step 4.3.7.1.4

Rewrite $- 1468$ as $- 1 (1468)$ .

$x = \frac{146 \pm \sqrt{- 1 \cdot 1468}}{2 \cdot 64}$

Step 4.3.7.1.5

Rewrite $\sqrt{- 1 (1468)}$ as $\sqrt{- 1} \cdot \sqrt{1468}$ .

$x = \frac{146 \pm \sqrt{- 1} \cdot \sqrt{1468}}{2 \cdot 64}$

Step 4.3.7.1.6

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

$x = \frac{146 \pm i \cdot \sqrt{1468}}{2 \cdot 64}$

Step 4.3.7.1.7

Rewrite $1468$ as $2^{2} \cdot 367$ .

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Step 4.3.7.1.7.1

Factor $4$ out of $1468$ .

$x = \frac{146 \pm i \cdot \sqrt{4 (367)}}{2 \cdot 64}$

Step 4.3.7.1.7.2

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

$x = \frac{146 \pm i \cdot \sqrt{2^{2} \cdot 367}}{2 \cdot 64}$

Step 4.3.7.1.8

Pull terms out from under the radical.

$x = \frac{146 \pm i \cdot (2 \sqrt{367})}{2 \cdot 64}$

Step 4.3.7.1.9

Move $2$ to the left of $i$ .

$x = \frac{146 \pm 2 i \sqrt{367}}{2 \cdot 64}$

Step 4.3.7.2

Multiply $2$ by $64$ .

$x = \frac{146 \pm 2 i \sqrt{367}}{128}$

Step 4.3.7.3

Simplify $\frac{146 \pm 2 i \sqrt{367}}{128}$ .

$x = \frac{73 \pm i \sqrt{367}}{64}$

Step 4.3.8

The final answer is the combination of both solutions.

$x = \frac{73 + i \sqrt{367}}{64}, \frac{73 - i \sqrt{367}}{64}$