Finite Math Examples

$103 + x^{2} = 47 + \frac{5}{2} x^{2}$

Step 1

Combine $\frac{5}{2}$ and $x^{2}$ .

$103 + x^{2} = 47 + \frac{5 x^{2}}{2}$

Step 2

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

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Step 2.1

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

$103 + x^{2} - \frac{5 x^{2}}{2} = 47$

Step 2.2

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

$103 + x^{2} \cdot \frac{2}{2} - \frac{5 x^{2}}{2} = 47$

Step 2.3

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

$103 + \frac{x^{2} \cdot 2}{2} - \frac{5 x^{2}}{2} = 47$

Step 2.4

Combine the numerators over the common denominator.

$103 + \frac{x^{2} \cdot 2 - 5 x^{2}}{2} = 47$

Step 2.5

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

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Step 2.5.1

Reorder $x^{2}$ and $2$ .

$103 + \frac{2 \cdot x^{2} - 5 x^{2}}{2} = 47$

Step 2.5.2

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

$103 + \frac{- 3 \cdot x^{2}}{2} = 47$

Step 2.6

Move the negative in front of the fraction.

$103 - \frac{3 x^{2}}{2} = 47$

Step 3

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

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Step 3.1

Subtract $103$ from both sides of the equation.

$- \frac{3 x^{2}}{2} = 47 - 103$

Step 3.2

Subtract $103$ from $47$ .

$- \frac{3 x^{2}}{2} = - 56$

Step 4

Multiply both sides of the equation by $- \frac{2}{3}$ .

$- \frac{2}{3} (- \frac{3 x^{2}}{2}) = - \frac{2}{3} \cdot - 56$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $- \frac{2}{3} (- \frac{3 x^{2}}{2})$ .

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Step 5.1.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$\frac{- 2}{3} (- \frac{3 x^{2}}{2}) = - \frac{2}{3} \cdot - 56$

Step 5.1.1.1.2

Move the leading negative in $- \frac{3 x^{2}}{2}$ into the numerator.

$\frac{- 2}{3} \cdot \frac{- 3 x^{2}}{2} = - \frac{2}{3} \cdot - 56$

Step 5.1.1.1.3

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{3} \cdot \frac{- 3 x^{2}}{2} = - \frac{2}{3} \cdot - 56$

Step 5.1.1.1.4

Cancel the common factor.

$\frac{2 \cdot - 1}{3} \cdot \frac{- 3 x^{2}}{2} = - \frac{2}{3} \cdot - 56$

Step 5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{3} (- 3 x^{2}) = - \frac{2}{3} \cdot - 56$

Step 5.1.1.2

Cancel the common factor of $3$ .

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Step 5.1.1.2.1

Factor $3$ out of $- 3 x^{2}$ .

$\frac{- 1}{3} (3 (- x^{2})) = - \frac{2}{3} \cdot - 56$

Step 5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{3} (3 (- x^{2})) = - \frac{2}{3} \cdot - 56$

Step 5.1.1.2.3

Rewrite the expression.

$- - x^{2} = - \frac{2}{3} \cdot - 56$

Step 5.1.1.3

Multiply.

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Step 5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x^{2} = - \frac{2}{3} \cdot - 56$

Step 5.1.1.3.2

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

$x^{2} = - \frac{2}{3} \cdot - 56$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $- \frac{2}{3} \cdot - 56$ .

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Step 5.2.1.1

Multiply $- 56$ by $- 1$ .

$x^{2} = 56 (\frac{2}{3})$

Step 5.2.1.2

Combine $56$ and $\frac{2}{3}$ .

$x^{2} = \frac{56 \cdot 2}{3}$

Step 5.2.1.3

Multiply $56$ by $2$ .

$x^{2} = \frac{112}{3}$

Step 6

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

$x = \pm \sqrt{\frac{112}{3}}$

Step 7

Simplify $\pm \sqrt{\frac{112}{3}}$ .

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Step 7.1

Rewrite $\sqrt{\frac{112}{3}}$ as $\frac{\sqrt{112}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{112}}{\sqrt{3}}$

Step 7.2

Simplify the numerator.

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Step 7.2.1

Rewrite $112$ as $4^{2} \cdot 7$ .

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Step 7.2.1.1

Factor $16$ out of $112$ .

$x = \pm \frac{\sqrt{16 (7)}}{\sqrt{3}}$

Step 7.2.1.2

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

$x = \pm \frac{\sqrt{4^{2} \cdot 7}}{\sqrt{3}}$

Step 7.2.2

Pull terms out from under the radical.

$x = \pm \frac{4 \sqrt{7}}{\sqrt{3}}$

Step 7.3

Multiply $\frac{4 \sqrt{7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{4 \sqrt{7}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 7.4

Combine and simplify the denominator.

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Step 7.4.1

Multiply $\frac{4 \sqrt{7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{4 \sqrt{7} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 7.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{4 \sqrt{7} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 7.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{4 \sqrt{7} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 7.4.4

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

$x = \pm \frac{4 \sqrt{7} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 7.4.5

Add $1$ and $1$ .

$x = \pm \frac{4 \sqrt{7} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 7.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 7.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm \frac{4 \sqrt{7} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 7.4.6.2

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

$x = \pm \frac{4 \sqrt{7} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 7.4.6.3

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

$x = \pm \frac{4 \sqrt{7} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 7.4.6.4

Cancel the common factor of $2$ .

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Step 7.4.6.4.1

Cancel the common factor.

$x = \pm \frac{4 \sqrt{7} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 7.4.6.4.2

Rewrite the expression.

$x = \pm \frac{4 \sqrt{7} \sqrt{3}}{3^{1}}$

Step 7.4.6.5

Evaluate the exponent.

$x = \pm \frac{4 \sqrt{7} \sqrt{3}}{3}$

Step 7.5

Simplify the numerator.

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Step 7.5.1

Combine using the product rule for radicals.

$x = \pm \frac{4 \sqrt{3 \cdot 7}}{3}$

Step 7.5.2

Multiply $3$ by $7$ .

$x = \pm \frac{4 \sqrt{21}}{3}$

Step 8

The complete solution is the result of both the positive and negative portions of the solution.

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Step 8.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{4 \sqrt{21}}{3}$

Step 8.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{4 \sqrt{21}}{3}$

Step 8.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{4 \sqrt{21}}{3}, - \frac{4 \sqrt{21}}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4 \sqrt{21}}{3}, - \frac{4 \sqrt{21}}{3}$

Decimal Form:

$x = 6.11010092 \dots, - 6.11010092 \dots$