Finite Math Examples

$z = \sqrt{1 - x^{2}} + \sqrt{y^{2} - 1}$

Step 1

Rewrite the equation as $\sqrt{1 - x^{2}} + \sqrt{y^{2} - 1} = z$ .

$\sqrt{1 - x^{2}} + \sqrt{y^{2} - 1} = z$

Step 2

Solve for $\sqrt{1 - x^{2}}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite $1$ as $1^{2}$ .

$\sqrt{1 - x^{2}} + \sqrt{y^{2} - 1^{2}} = z$

Step 2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 1$ .

$\sqrt{1 - x^{2}} + \sqrt{(y + 1) (y - 1)} = z$

Step 2.2

Subtract $\sqrt{(y + 1) (y - 1)}$ from both sides of the equation.

$\sqrt{1 - x^{2}} = z - \sqrt{(y + 1) (y - 1)}$

Step 3

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

${\sqrt{1 - x^{2}}}^{2} = {(z - \sqrt{(y + 1) (y - 1)})}^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

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

${({(1 - x^{2})}^{\frac{1}{2}})}^{2} = {(z - \sqrt{(y + 1) (y - 1)})}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

${(1 - x^{2})}^{\frac{1}{2} \cdot 2} = {(z - \sqrt{(y + 1) (y - 1)})}^{2}$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

${(1 - x^{2})}^{\frac{1}{2} \cdot 2} = {(z - \sqrt{(y + 1) (y - 1)})}^{2}$

Step 4.2.1.1.2.2

Rewrite the expression.

${(1 - x^{2})}^{1} = {(z - \sqrt{(y + 1) (y - 1)})}^{2}$

Step 4.2.1.2

Simplify.

$1 - x^{2} = {(z - \sqrt{(y + 1) (y - 1)})}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(z - \sqrt{(y + 1) (y - 1)})}^{2}$ .

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Step 4.3.1.1

Rewrite ${(z - \sqrt{(y + 1) (y - 1)})}^{2}$ as $(z - \sqrt{(y + 1) (y - 1)}) (z - \sqrt{(y + 1) (y - 1)})$ .

$1 - x^{2} = (z - \sqrt{(y + 1) (y - 1)}) (z - \sqrt{(y + 1) (y - 1)})$

Step 4.3.1.2

Expand $(z - \sqrt{(y + 1) (y - 1)}) (z - \sqrt{(y + 1) (y - 1)})$ using the FOIL Method.

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Step 4.3.1.2.1

Apply the distributive property.

$1 - x^{2} = z (z - \sqrt{(y + 1) (y - 1)}) - \sqrt{(y + 1) (y - 1)} (z - \sqrt{(y + 1) (y - 1)})$

Step 4.3.1.2.2

Apply the distributive property.

$1 - x^{2} = z \cdot z + z (- \sqrt{(y + 1) (y - 1)}) - \sqrt{(y + 1) (y - 1)} (z - \sqrt{(y + 1) (y - 1)})$

Step 4.3.1.2.3

Apply the distributive property.

$1 - x^{2} = z \cdot z + z (- \sqrt{(y + 1) (y - 1)}) - \sqrt{(y + 1) (y - 1)} z - \sqrt{(y + 1) (y - 1)} (- \sqrt{(y + 1) (y - 1)})$

Step 4.3.1.3

Simplify and combine like terms.

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Step 4.3.1.3.1

Simplify each term.

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Step 4.3.1.3.1.1

Multiply $z$ by $z$ .

$1 - x^{2} = z^{2} + z (- \sqrt{(y + 1) (y - 1)}) - \sqrt{(y + 1) (y - 1)} z - \sqrt{(y + 1) (y - 1)} (- \sqrt{(y + 1) (y - 1)})$

Step 4.3.1.3.1.2

Rewrite using the commutative property of multiplication.

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z - \sqrt{(y + 1) (y - 1)} (- \sqrt{(y + 1) (y - 1)})$

Step 4.3.1.3.1.3

Multiply $- \sqrt{(y + 1) (y - 1)} (- \sqrt{(y + 1) (y - 1)})$ .

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Step 4.3.1.3.1.3.1

Multiply $- 1$ by $- 1$ .

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + 1 \sqrt{(y + 1) (y - 1)} \sqrt{(y + 1) (y - 1)}$

Step 4.3.1.3.1.3.2

Multiply $\sqrt{(y + 1) (y - 1)}$ by $1$ .

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + \sqrt{(y + 1) (y - 1)} \sqrt{(y + 1) (y - 1)}$

Step 4.3.1.3.1.3.3

Raise $\sqrt{(y + 1) (y - 1)}$ to the power of $1$ .

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + {\sqrt{(y + 1) (y - 1)}}^{1} \sqrt{(y + 1) (y - 1)}$

Step 4.3.1.3.1.3.4

Raise $\sqrt{(y + 1) (y - 1)}$ to the power of $1$ .

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + {\sqrt{(y + 1) (y - 1)}}^{1} {\sqrt{(y + 1) (y - 1)}}^{1}$

Step 4.3.1.3.1.3.5

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

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + {\sqrt{(y + 1) (y - 1)}}^{1 + 1}$

Step 4.3.1.3.1.3.6

Add $1$ and $1$ .

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + {\sqrt{(y + 1) (y - 1)}}^{2}$

Step 4.3.1.3.1.4

Rewrite ${\sqrt{(y + 1) (y - 1)}}^{2}$ as $(y + 1) (y - 1)$ .

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Step 4.3.1.3.1.4.1

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

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + {({((y + 1) (y - 1))}^{\frac{1}{2}})}^{2}$

Step 4.3.1.3.1.4.2

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

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + {((y + 1) (y - 1))}^{\frac{1}{2} \cdot 2}$

Step 4.3.1.3.1.4.3

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

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + {((y + 1) (y - 1))}^{\frac{2}{2}}$

Step 4.3.1.3.1.4.4

Cancel the common factor of $2$ .

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Step 4.3.1.3.1.4.4.1

Cancel the common factor.

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + {((y + 1) (y - 1))}^{\frac{2}{2}}$

Step 4.3.1.3.1.4.4.2

Rewrite the expression.

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + {((y + 1) (y - 1))}^{1}$

Step 4.3.1.3.1.4.5

Simplify.

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + (y + 1) (y - 1)$

Step 4.3.1.3.1.5

Expand $(y + 1) (y - 1)$ using the FOIL Method.

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Step 4.3.1.3.1.5.1

Apply the distributive property.

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + y (y - 1) + 1 (y - 1)$

Step 4.3.1.3.1.5.2

Apply the distributive property.

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + y \cdot y + y \cdot - 1 + 1 (y - 1)$

Step 4.3.1.3.1.5.3

Apply the distributive property.

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + y \cdot y + y \cdot - 1 + 1 y + 1 \cdot - 1$

Step 4.3.1.3.1.6

Simplify and combine like terms.

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Step 4.3.1.3.1.6.1

Simplify each term.

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Step 4.3.1.3.1.6.1.1

Multiply $y$ by $y$ .

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + y^{2} + y \cdot - 1 + 1 y + 1 \cdot - 1$

Step 4.3.1.3.1.6.1.2

Move $- 1$ to the left of $y$ .

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + y^{2} - 1 \cdot y + 1 y + 1 \cdot - 1$

Step 4.3.1.3.1.6.1.3

Rewrite $- 1 y$ as $- y$ .

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + y^{2} - y + 1 y + 1 \cdot - 1$

Step 4.3.1.3.1.6.1.4

Multiply $y$ by $1$ .

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + y^{2} - y + y + 1 \cdot - 1$

Step 4.3.1.3.1.6.1.5

Multiply $- 1$ by $1$ .

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + y^{2} - y + y - 1$

Step 4.3.1.3.1.6.2

Add $- y$ and $y$ .

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + y^{2} + 0 - 1$

Step 4.3.1.3.1.6.3

Add $y^{2}$ and $0$ .

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - \sqrt{(y + 1) (y - 1)} z + y^{2} - 1$

Step 4.3.1.3.2

Reorder the factors of $- \sqrt{(y + 1) (y - 1)} z$ .

$1 - x^{2} = z^{2} - z \sqrt{(y + 1) (y - 1)} - z \sqrt{(y + 1) (y - 1)} + y^{2} - 1$

Step 4.3.1.3.3

Subtract $z \sqrt{(y + 1) (y - 1)}$ from $- z \sqrt{(y + 1) (y - 1)}$ .

$1 - x^{2} = z^{2} - 2 z \sqrt{(y + 1) (y - 1)} + y^{2} - 1$

Step 5

Solve for $x$ .

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Step 5.1

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

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Step 5.1.1

Subtract $1$ from both sides of the equation.

$- x^{2} = z^{2} - 2 z \sqrt{(y + 1) (y - 1)} + y^{2} - 1 - 1$

Step 5.1.2

Subtract $1$ from $- 1$ .

$- x^{2} = z^{2} - 2 z \sqrt{(y + 1) (y - 1)} + y^{2} - 2$

Step 5.2

Divide each term in $- x^{2} = z^{2} - 2 z \sqrt{(y + 1) (y - 1)} + y^{2} - 2$ by $- 1$ and simplify.

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Step 5.2.1

Divide each term in $- x^{2} = z^{2} - 2 z \sqrt{(y + 1) (y - 1)} + y^{2} - 2$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{z^{2}}{- 1} + \frac{- 2 z \sqrt{(y + 1) (y - 1)}}{- 1} + \frac{y^{2}}{- 1} + \frac{- 2}{- 1}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{z^{2}}{- 1} + \frac{- 2 z \sqrt{(y + 1) (y - 1)}}{- 1} + \frac{y^{2}}{- 1} + \frac{- 2}{- 1}$

Step 5.2.2.2

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

$x^{2} = \frac{z^{2}}{- 1} + \frac{- 2 z \sqrt{(y + 1) (y - 1)}}{- 1} + \frac{y^{2}}{- 1} + \frac{- 2}{- 1}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Move the negative one from the denominator of $\frac{z^{2}}{- 1}$ .

$x^{2} = - 1 \cdot z^{2} + \frac{- 2 z \sqrt{(y + 1) (y - 1)}}{- 1} + \frac{y^{2}}{- 1} + \frac{- 2}{- 1}$

Step 5.2.3.1.2

Rewrite $- 1 \cdot z^{2}$ as $- z^{2}$ .

$x^{2} = - z^{2} + \frac{- 2 z \sqrt{(y + 1) (y - 1)}}{- 1} + \frac{y^{2}}{- 1} + \frac{- 2}{- 1}$

Step 5.2.3.1.3

Move the negative one from the denominator of $\frac{- 2 z \sqrt{(y + 1) (y - 1)}}{- 1}$ .

$x^{2} = - z^{2} - 1 \cdot (- 2 z \sqrt{(y + 1) (y - 1)}) + \frac{y^{2}}{- 1} + \frac{- 2}{- 1}$

Step 5.2.3.1.4

Rewrite $- 1 \cdot (- 2 z \sqrt{(y + 1) (y - 1)})$ as $- (- 2 z \sqrt{(y + 1) (y - 1)})$ .

$x^{2} = - z^{2} - (- 2 z \sqrt{(y + 1) (y - 1)}) + \frac{y^{2}}{- 1} + \frac{- 2}{- 1}$

Step 5.2.3.1.5

Multiply $- 2$ by $- 1$ .

$x^{2} = - z^{2} + 2 (z \sqrt{(y + 1) (y - 1)}) + \frac{y^{2}}{- 1} + \frac{- 2}{- 1}$

Step 5.2.3.1.6

Move the negative one from the denominator of $\frac{y^{2}}{- 1}$ .

$x^{2} = - z^{2} + 2 z \sqrt{(y + 1) (y - 1)} - 1 \cdot y^{2} + \frac{- 2}{- 1}$

Step 5.2.3.1.7

Rewrite $- 1 \cdot y^{2}$ as $- y^{2}$ .

$x^{2} = - z^{2} + 2 z \sqrt{(y + 1) (y - 1)} - y^{2} + \frac{- 2}{- 1}$

Step 5.2.3.1.8

Divide $- 2$ by $- 1$ .

$x^{2} = - z^{2} + 2 z \sqrt{(y + 1) (y - 1)} - y^{2} + 2$

Step 5.3

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

$x = \pm \sqrt{- z^{2} + 2 z \sqrt{(y + 1) (y - 1)} - y^{2} + 2}$

Step 5.4

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

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Step 5.4.1

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

$x = \sqrt{- z^{2} + 2 z \sqrt{(y + 1) (y - 1)} - y^{2} + 2}$

Step 5.4.2

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

$x = - \sqrt{- z^{2} + 2 z \sqrt{(y + 1) (y - 1)} - y^{2} + 2}$

Step 5.4.3

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

$x = \sqrt{- z^{2} + 2 z \sqrt{(y + 1) (y - 1)} - y^{2} + 2}$

$x = - \sqrt{- z^{2} + 2 z \sqrt{(y + 1) (y - 1)} - y^{2} + 2}$

$x = \sqrt{- z^{2} + 2 z \sqrt{(y + 1) (y - 1)} - y^{2} + 2}$

$x = - \sqrt{- z^{2} + 2 z \sqrt{(y + 1) (y - 1)} - y^{2} + 2}$

$x = \sqrt{- z^{2} + 2 z \sqrt{(y + 1) (y - 1)} - y^{2} + 2}$

$x = - \sqrt{- z^{2} + 2 z \sqrt{(y + 1) (y - 1)} - y^{2} + 2}$