Basic Math Examples

$\frac{g + h}{3} = \frac{3}{g - h}$

Step 1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$(g + h) (g - h) = 3 \cdot 3$

Step 2

Solve the equation for $h$ .

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Step 2.1

Simplify $(g + h) (g - h)$ .

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Step 2.1.1

Expand $(g + h) (g - h)$ using the FOIL Method.

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Step 2.1.1.1

Apply the distributive property.

$g (g - h) + h (g - h) = 3 \cdot 3$

Step 2.1.1.2

Apply the distributive property.

$g \cdot g + g (- h) + h (g - h) = 3 \cdot 3$

Step 2.1.1.3

Apply the distributive property.

$g \cdot g + g (- h) + h g + h (- h) = 3 \cdot 3$

Step 2.1.2

Simplify terms.

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Step 2.1.2.1

Combine the opposite terms in $g \cdot g + g (- h) + h g + h (- h)$ .

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Step 2.1.2.1.1

Reorder the factors in the terms $g (- h)$ and $h g$ .

$g \cdot g - g h + g h + h (- h) = 3 \cdot 3$

Step 2.1.2.1.2

Add $- g h$ and $g h$ .

$g \cdot g + 0 + h (- h) = 3 \cdot 3$

Step 2.1.2.1.3

Add $g \cdot g$ and $0$ .

$g \cdot g + h (- h) = 3 \cdot 3$

Step 2.1.2.2

Simplify each term.

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Step 2.1.2.2.1

Multiply $g$ by $g$ .

$g^{2} + h (- h) = 3 \cdot 3$

Step 2.1.2.2.2

Rewrite using the commutative property of multiplication.

$g^{2} - h \cdot h = 3 \cdot 3$

Step 2.1.2.2.3

Multiply $h$ by $h$ by adding the exponents.

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Step 2.1.2.2.3.1

Move $h$ .

$g^{2} - (h \cdot h) = 3 \cdot 3$

Step 2.1.2.2.3.2

Multiply $h$ by $h$ .

$g^{2} - h^{2} = 3 \cdot 3$

Step 2.2

Multiply $3$ by $3$ .

$g^{2} - h^{2} = 9$

Step 2.3

Subtract $g^{2}$ from both sides of the equation.

$- h^{2} = 9 - g^{2}$

Step 2.4

Divide each term in $- h^{2} = 9 - g^{2}$ by $- 1$ and simplify.

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Step 2.4.1

Divide each term in $- h^{2} = 9 - g^{2}$ by $- 1$ .

$\frac{- h^{2}}{- 1} = \frac{9}{- 1} + \frac{- g^{2}}{- 1}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Dividing two negative values results in a positive value.

$\frac{h^{2}}{1} = \frac{9}{- 1} + \frac{- g^{2}}{- 1}$

Step 2.4.2.2

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

$h^{2} = \frac{9}{- 1} + \frac{- g^{2}}{- 1}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

Divide $9$ by $- 1$ .

$h^{2} = - 9 + \frac{- g^{2}}{- 1}$

Step 2.4.3.1.2

Dividing two negative values results in a positive value.

$h^{2} = - 9 + \frac{g^{2}}{1}$

Step 2.4.3.1.3

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

$h^{2} = - 9 + g^{2}$

Step 2.5

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

$h = \pm \sqrt{- 9 + g^{2}}$

Step 2.6

Simplify $\pm \sqrt{- 9 + g^{2}}$ .

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Step 2.6.1

Simplify the expression.

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Step 2.6.1.1

Rewrite $9$ as $3^{2}$ .

$h = \pm \sqrt{- 3^{2} + g^{2}}$

Step 2.6.1.2

Reorder $- 3^{2}$ and $g^{2}$ .

$h = \pm \sqrt{g^{2} - 3^{2}}$

Step 2.6.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = g$ and $b = 3$ .

$h = \pm \sqrt{(g + 3) (g - 3)}$

Step 2.7

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

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Step 2.7.1

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

$h = \sqrt{(g + 3) (g - 3)}$

Step 2.7.2

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

$h = - \sqrt{(g + 3) (g - 3)}$

Step 2.7.3

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

$h = \sqrt{(g + 3) (g - 3)}$

$h = - \sqrt{(g + 3) (g - 3)}$

$h = \sqrt{(g + 3) (g - 3)}$

$h = - \sqrt{(g + 3) (g - 3)}$

$h = \sqrt{(g + 3) (g - 3)}$

$h = - \sqrt{(g + 3) (g - 3)}$