Finite Math Examples

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

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

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

$2 x + \sqrt{\frac{x - 8 + 4}{2}} - 2 = 9 (\frac{1}{6}) - 3 x$

Step 1.1.2

Cancel the common factor of $3$ .

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Step 1.1.2.1

Factor $3$ out of $9$ .

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

Step 1.1.2.2

Factor $3$ out of $6$ .

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

Step 1.1.2.3

Cancel the common factor.

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

Step 1.1.2.4

Rewrite the expression.

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

Step 1.1.3

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

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

Step 1.2

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

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Step 1.2.1

Subtract $2 x$ from both sides of the equation.

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

Step 1.2.2

Add $2$ to both sides of the equation.

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Combine the numerators over the common denominator.

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

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $2$ by $2$ .

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

Step 1.2.6.2

Add $3$ and $4$ .

$\sqrt{\frac{x - 8 + 4}{2}} = - 3 x - 2 x + \frac{7}{2}$

Step 1.2.7

Subtract $2 x$ from $- 3 x$ .

$\sqrt{\frac{x - 8 + 4}{2}} = - 5 x + \frac{7}{2}$

Step 2

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

${\sqrt{\frac{x - 8 + 4}{2}}}^{2} = {(- 5 x + \frac{7}{2})}^{2}$

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

${({(\frac{x - 8 + 4}{2})}^{\frac{1}{2}})}^{2} = {(- 5 x + \frac{7}{2})}^{2}$

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, ${(a^{m})}^{n} = a^{m n}$ .

${(\frac{x - 8 + 4}{2})}^{\frac{1}{2} \cdot 2} = {(- 5 x + \frac{7}{2})}^{2}$

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.

${(\frac{x - 8 + 4}{2})}^{\frac{1}{2} \cdot 2} = {(- 5 x + \frac{7}{2})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(\frac{x - 8 + 4}{2})}^{1} = {(- 5 x + \frac{7}{2})}^{2}$

Step 3.2.1.2

Add $- 8$ and $4$ .

${(\frac{x - 4}{2})}^{1} = {(- 5 x + \frac{7}{2})}^{2}$

Step 3.2.1.3

Simplify.

$\frac{x - 4}{2} = {(- 5 x + \frac{7}{2})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(- 5 x + \frac{7}{2})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(- 5 x + \frac{7}{2})}^{2}$ as $(- 5 x + \frac{7}{2}) (- 5 x + \frac{7}{2})$ .

$\frac{x - 4}{2} = (- 5 x + \frac{7}{2}) (- 5 x + \frac{7}{2})$

Step 3.3.1.2

Expand $(- 5 x + \frac{7}{2}) (- 5 x + \frac{7}{2})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$\frac{x - 4}{2} = - 5 x (- 5 x + \frac{7}{2}) + \frac{7}{2} (- 5 x + \frac{7}{2})$

Step 3.3.1.2.2

Apply the distributive property.

$\frac{x - 4}{2} = - 5 x (- 5 x) - 5 x \frac{7}{2} + \frac{7}{2} (- 5 x + \frac{7}{2})$

Step 3.3.1.2.3

Apply the distributive property.

$\frac{x - 4}{2} = - 5 x (- 5 x) - 5 x \frac{7}{2} + \frac{7}{2} (- 5 x) + \frac{7}{2} \cdot \frac{7}{2}$

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

Rewrite using the commutative property of multiplication.

$\frac{x - 4}{2} = - 5 \cdot - 5 x \cdot x - 5 x \frac{7}{2} + \frac{7}{2} (- 5 x) + \frac{7}{2} \cdot \frac{7}{2}$

Step 3.3.1.3.1.2

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

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Step 3.3.1.3.1.2.1

Move $x$ .

$\frac{x - 4}{2} = - 5 \cdot - 5 (x \cdot x) - 5 x \frac{7}{2} + \frac{7}{2} (- 5 x) + \frac{7}{2} \cdot \frac{7}{2}$

Step 3.3.1.3.1.2.2

Multiply $x$ by $x$ .

$\frac{x - 4}{2} = - 5 \cdot - 5 x^{2} - 5 x \frac{7}{2} + \frac{7}{2} (- 5 x) + \frac{7}{2} \cdot \frac{7}{2}$

Step 3.3.1.3.1.3

Multiply $- 5$ by $- 5$ .

$\frac{x - 4}{2} = 25 x^{2} - 5 x \frac{7}{2} + \frac{7}{2} (- 5 x) + \frac{7}{2} \cdot \frac{7}{2}$

Step 3.3.1.3.1.4

Multiply $- 5 x \frac{7}{2}$ .

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Step 3.3.1.3.1.4.1

Combine $- 5$ and $\frac{7}{2}$ .

$\frac{x - 4}{2} = 25 x^{2} + x \frac{- 5 \cdot 7}{2} + \frac{7}{2} (- 5 x) + \frac{7}{2} \cdot \frac{7}{2}$

Step 3.3.1.3.1.4.2

Multiply $- 5$ by $7$ .

$\frac{x - 4}{2} = 25 x^{2} + x \frac{- 35}{2} + \frac{7}{2} (- 5 x) + \frac{7}{2} \cdot \frac{7}{2}$

Step 3.3.1.3.1.4.3

Combine $x$ and $\frac{- 35}{2}$ .

$\frac{x - 4}{2} = 25 x^{2} + \frac{x \cdot - 35}{2} + \frac{7}{2} (- 5 x) + \frac{7}{2} \cdot \frac{7}{2}$

Step 3.3.1.3.1.5

Move $- 35$ to the left of $x$ .

$\frac{x - 4}{2} = 25 x^{2} + \frac{- 35 \cdot x}{2} + \frac{7}{2} (- 5 x) + \frac{7}{2} \cdot \frac{7}{2}$

Step 3.3.1.3.1.6

Move the negative in front of the fraction.

$\frac{x - 4}{2} = 25 x^{2} - \frac{35 \cdot x}{2} + \frac{7}{2} (- 5 x) + \frac{7}{2} \cdot \frac{7}{2}$

Step 3.3.1.3.1.7

Multiply $\frac{7}{2} (- 5 x)$ .

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Step 3.3.1.3.1.7.1

Combine $- 5$ and $\frac{7}{2}$ .

$\frac{x - 4}{2} = 25 x^{2} - \frac{35 x}{2} + \frac{- 5 \cdot 7}{2} x + \frac{7}{2} \cdot \frac{7}{2}$

Step 3.3.1.3.1.7.2

Multiply $- 5$ by $7$ .

$\frac{x - 4}{2} = 25 x^{2} - \frac{35 x}{2} + \frac{- 35}{2} x + \frac{7}{2} \cdot \frac{7}{2}$

Step 3.3.1.3.1.7.3

Combine $\frac{- 35}{2}$ and $x$ .

$\frac{x - 4}{2} = 25 x^{2} - \frac{35 x}{2} + \frac{- 35 x}{2} + \frac{7}{2} \cdot \frac{7}{2}$

Step 3.3.1.3.1.8

Move the negative in front of the fraction.

$\frac{x - 4}{2} = 25 x^{2} - \frac{35 x}{2} - \frac{35 x}{2} + \frac{7}{2} \cdot \frac{7}{2}$

Step 3.3.1.3.1.9

Multiply $\frac{7}{2} \cdot \frac{7}{2}$ .

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Step 3.3.1.3.1.9.1

Multiply $\frac{7}{2}$ by $\frac{7}{2}$ .

$\frac{x - 4}{2} = 25 x^{2} - \frac{35 x}{2} - \frac{35 x}{2} + \frac{7 \cdot 7}{2 \cdot 2}$

Step 3.3.1.3.1.9.2

Multiply $7$ by $7$ .

$\frac{x - 4}{2} = 25 x^{2} - \frac{35 x}{2} - \frac{35 x}{2} + \frac{49}{2 \cdot 2}$

Step 3.3.1.3.1.9.3

Multiply $2$ by $2$ .

$\frac{x - 4}{2} = 25 x^{2} - \frac{35 x}{2} - \frac{35 x}{2} + \frac{49}{4}$

Step 3.3.1.3.2

Subtract $\frac{35 x}{2}$ from $- \frac{35 x}{2}$ .

$\frac{x - 4}{2} = 25 x^{2} - 2 \frac{35 x}{2} + \frac{49}{4}$

Step 3.3.1.4

Simplify each term.

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Step 3.3.1.4.1

Cancel the common factor of $2$ .

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Step 3.3.1.4.1.1

Factor $2$ out of $- 2$ .

$\frac{x - 4}{2} = 25 x^{2} + 2 (- 1) \frac{35 x}{2} + \frac{49}{4}$

Step 3.3.1.4.1.2

Cancel the common factor.

$\frac{x - 4}{2} = 25 x^{2} + 2 \cdot - 1 \frac{35 x}{2} + \frac{49}{4}$

Step 3.3.1.4.1.3

Rewrite the expression.

$\frac{x - 4}{2} = 25 x^{2} - 1 (35 x) + \frac{49}{4}$

Step 3.3.1.4.2

Multiply $35$ by $- 1$ .

$\frac{x - 4}{2} = 25 x^{2} - 35 x + \frac{49}{4}$

Step 4

Solve for $x$ .

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Step 4.1

Multiply both sides by $2$ .

$\frac{x - 4}{2} \cdot 2 = (25 x^{2} - 35 x + \frac{49}{4}) \cdot 2$

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.

$\frac{x - 4}{2} \cdot 2 = (25 x^{2} - 35 x + \frac{49}{4}) \cdot 2$

Step 4.2.1.1.2

Rewrite the expression.

$x - 4 = (25 x^{2} - 35 x + \frac{49}{4}) \cdot 2$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $(25 x^{2} - 35 x + \frac{49}{4}) \cdot 2$ .

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Step 4.2.2.1.1

Apply the distributive property.

$x - 4 = 25 x^{2} \cdot 2 - 35 x \cdot 2 + \frac{49}{4} \cdot 2$

Step 4.2.2.1.2

Simplify.

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Step 4.2.2.1.2.1

Multiply $2$ by $25$ .

$x - 4 = 50 x^{2} - 35 x \cdot 2 + \frac{49}{4} \cdot 2$

Step 4.2.2.1.2.2

Multiply $2$ by $- 35$ .

$x - 4 = 50 x^{2} - 70 x + \frac{49}{4} \cdot 2$

Step 4.2.2.1.2.3

Cancel the common factor of $2$ .

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Step 4.2.2.1.2.3.1

Factor $2$ out of $4$ .

$x - 4 = 50 x^{2} - 70 x + \frac{49}{2 (2)} \cdot 2$

Step 4.2.2.1.2.3.2

Cancel the common factor.

$x - 4 = 50 x^{2} - 70 x + \frac{49}{2 \cdot 2} \cdot 2$

Step 4.2.2.1.2.3.3

Rewrite the expression.

$x - 4 = 50 x^{2} - 70 x + \frac{49}{2}$

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.

$50 x^{2} - 70 x + \frac{49}{2} = x - 4$

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.

$50 x^{2} - 70 x + \frac{49}{2} - x = - 4$

Step 4.3.2.2

Subtract $x$ from $- 70 x$ .

$50 x^{2} - 71 x + \frac{49}{2} = - 4$

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.

$50 x^{2} - 71 x + \frac{49}{2} + 4 = 0$

Step 4.3.3.2

Simplify $50 x^{2} - 71 x + \frac{49}{2} + 4$ .

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Step 4.3.3.2.1

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

$50 x^{2} - 71 x + \frac{49}{2} + 4 \cdot \frac{2}{2} = 0$

Step 4.3.3.2.2

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

$50 x^{2} - 71 x + \frac{49}{2} + \frac{4 \cdot 2}{2} = 0$

Step 4.3.3.2.3

Combine the numerators over the common denominator.

$50 x^{2} - 71 x + \frac{49 + 4 \cdot 2}{2} = 0$

Step 4.3.3.2.4

Simplify the numerator.

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Step 4.3.3.2.4.1

Multiply $4$ by $2$ .

$50 x^{2} - 71 x + \frac{49 + 8}{2} = 0$

Step 4.3.3.2.4.2

Add $49$ and $8$ .

$50 x^{2} - 71 x + \frac{57}{2} = 0$

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.

$2 (50 x^{2}) + 2 (- 71 x) + 2 (\frac{57}{2}) = 0$

Step 4.3.4.2

Simplify.

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Step 4.3.4.2.1

Multiply $50$ by $2$ .

$100 x^{2} + 2 (- 71 x) + 2 (\frac{57}{2}) = 0$

Step 4.3.4.2.2

Multiply $- 71$ by $2$ .

$100 x^{2} - 142 x + 2 (\frac{57}{2}) = 0$

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.

$100 x^{2} - 142 x + 2 (\frac{57}{2}) = 0$

Step 4.3.4.2.3.2

Rewrite the expression.

$100 x^{2} - 142 x + 57 = 0$

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 $a = 100$ , $b = - 142$ , and $c = 57$ into the quadratic formula and solve for $x$ .

$\frac{142 \pm \sqrt{{(- 142)}^{2} - 4 \cdot (100 \cdot 57)}}{2 \cdot 100}$

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 $- 142$ to the power of $2$ .

$x = \frac{142 \pm \sqrt{20164 - 4 \cdot 100 \cdot 57}}{2 \cdot 100}$

Step 4.3.7.1.2

Multiply $- 4 \cdot 100 \cdot 57$ .

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Step 4.3.7.1.2.1

Multiply $- 4$ by $100$ .

$x = \frac{142 \pm \sqrt{20164 - 400 \cdot 57}}{2 \cdot 100}$

Step 4.3.7.1.2.2

Multiply $- 400$ by $57$ .

$x = \frac{142 \pm \sqrt{20164 - 22800}}{2 \cdot 100}$

Step 4.3.7.1.3

Subtract $22800$ from $20164$ .

$x = \frac{142 \pm \sqrt{- 2636}}{2 \cdot 100}$

Step 4.3.7.1.4

Rewrite $- 2636$ as $- 1 (2636)$ .

$x = \frac{142 \pm \sqrt{- 1 \cdot 2636}}{2 \cdot 100}$

Step 4.3.7.1.5

Rewrite $\sqrt{- 1 (2636)}$ as $\sqrt{- 1} \cdot \sqrt{2636}$ .

$x = \frac{142 \pm \sqrt{- 1} \cdot \sqrt{2636}}{2 \cdot 100}$

Step 4.3.7.1.6

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

$x = \frac{142 \pm i \cdot \sqrt{2636}}{2 \cdot 100}$

Step 4.3.7.1.7

Rewrite $2636$ as $2^{2} \cdot 659$ .

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Step 4.3.7.1.7.1

Factor $4$ out of $2636$ .

$x = \frac{142 \pm i \cdot \sqrt{4 (659)}}{2 \cdot 100}$

Step 4.3.7.1.7.2

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

$x = \frac{142 \pm i \cdot \sqrt{2^{2} \cdot 659}}{2 \cdot 100}$

Step 4.3.7.1.8

Pull terms out from under the radical.

$x = \frac{142 \pm i \cdot (2 \sqrt{659})}{2 \cdot 100}$

Step 4.3.7.1.9

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

$x = \frac{142 \pm 2 i \sqrt{659}}{2 \cdot 100}$

Step 4.3.7.2

Multiply $2$ by $100$ .

$x = \frac{142 \pm 2 i \sqrt{659}}{200}$

Step 4.3.7.3

Simplify $\frac{142 \pm 2 i \sqrt{659}}{200}$ .

$x = \frac{71 \pm i \sqrt{659}}{100}$

Step 4.3.8

The final answer is the combination of both solutions.

$x = \frac{71 + i \sqrt{659}}{100}, \frac{71 - i \sqrt{659}}{100}$