Calculus Examples

$y = \sqrt{x - 3} - \sqrt{x + 3}$

Step 1

Rewrite the equation as $\sqrt{x - 3} - \sqrt{x + 3} = y$ .

$\sqrt{x - 3} - \sqrt{x + 3} = y$

Step 2

Add $\sqrt{x + 3}$ to both sides of the equation.

$\sqrt{x - 3} = y + \sqrt{x + 3}$

Step 3

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

${\sqrt{x - 3}}^{2} = {(y + \sqrt{x + 3})}^{2}$

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 $\sqrt{x - 3}$ as ${(x - 3)}^{\frac{1}{2}}$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

Multiply the exponents in ${({(x - 3)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1.1

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

${(x - 3)}^{\frac{1}{2} \cdot 2} = {(y + \sqrt{x + 3})}^{2}$

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.

${(x - 3)}^{\frac{1}{2} \cdot 2} = {(y + \sqrt{x + 3})}^{2}$

Step 4.2.1.1.2.2

Rewrite the expression.

${(x - 3)}^{1} = {(y + \sqrt{x + 3})}^{2}$

Step 4.2.1.2

Simplify.

$x - 3 = {(y + \sqrt{x + 3})}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(y + \sqrt{x + 3})}^{2}$ .

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Step 4.3.1.1

Rewrite ${(y + \sqrt{x + 3})}^{2}$ as $(y + \sqrt{x + 3}) (y + \sqrt{x + 3})$ .

$x - 3 = (y + \sqrt{x + 3}) (y + \sqrt{x + 3})$

Step 4.3.1.2

Expand $(y + \sqrt{x + 3}) (y + \sqrt{x + 3})$ using the FOIL Method.

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Step 4.3.1.2.1

Apply the distributive property.

$x - 3 = y (y + \sqrt{x + 3}) + \sqrt{x + 3} (y + \sqrt{x + 3})$

Step 4.3.1.2.2

Apply the distributive property.

$x - 3 = y \cdot y + y \sqrt{x + 3} + \sqrt{x + 3} (y + \sqrt{x + 3})$

Step 4.3.1.2.3

Apply the distributive property.

$x - 3 = y \cdot y + y \sqrt{x + 3} + \sqrt{x + 3} y + \sqrt{x + 3} \sqrt{x + 3}$

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 $y$ by $y$ .

$x - 3 = y^{2} + y \sqrt{x + 3} + \sqrt{x + 3} y + \sqrt{x + 3} \sqrt{x + 3}$

Step 4.3.1.3.1.2

Multiply $\sqrt{x + 3} \sqrt{x + 3}$ .

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Step 4.3.1.3.1.2.1

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

$x - 3 = y^{2} + y \sqrt{x + 3} + \sqrt{x + 3} y + {\sqrt{x + 3}}^{1} \sqrt{x + 3}$

Step 4.3.1.3.1.2.2

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

$x - 3 = y^{2} + y \sqrt{x + 3} + \sqrt{x + 3} y + {\sqrt{x + 3}}^{1} {\sqrt{x + 3}}^{1}$

Step 4.3.1.3.1.2.3

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

$x - 3 = y^{2} + y \sqrt{x + 3} + \sqrt{x + 3} y + {\sqrt{x + 3}}^{1 + 1}$

Step 4.3.1.3.1.2.4

Add $1$ and $1$ .

$x - 3 = y^{2} + y \sqrt{x + 3} + \sqrt{x + 3} y + {\sqrt{x + 3}}^{2}$

Step 4.3.1.3.1.3

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

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Step 4.3.1.3.1.3.1

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

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

Step 4.3.1.3.1.3.2

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

$x - 3 = y^{2} + y \sqrt{x + 3} + \sqrt{x + 3} y + {(x + 3)}^{\frac{1}{2} \cdot 2}$

Step 4.3.1.3.1.3.3

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

$x - 3 = y^{2} + y \sqrt{x + 3} + \sqrt{x + 3} y + {(x + 3)}^{\frac{2}{2}}$

Step 4.3.1.3.1.3.4

Cancel the common factor of $2$ .

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Step 4.3.1.3.1.3.4.1

Cancel the common factor.

$x - 3 = y^{2} + y \sqrt{x + 3} + \sqrt{x + 3} y + {(x + 3)}^{\frac{2}{2}}$

Step 4.3.1.3.1.3.4.2

Rewrite the expression.

$x - 3 = y^{2} + y \sqrt{x + 3} + \sqrt{x + 3} y + {(x + 3)}^{1}$

Step 4.3.1.3.1.3.5

Simplify.

$x - 3 = y^{2} + y \sqrt{x + 3} + \sqrt{x + 3} y + x + 3$

Step 4.3.1.3.2

Reorder the factors of $\sqrt{x + 3} y$ .

$x - 3 = y^{2} + y \sqrt{x + 3} + y \sqrt{x + 3} + x + 3$

Step 4.3.1.3.3

Add $y \sqrt{x + 3}$ and $y \sqrt{x + 3}$ .

$x - 3 = y^{2} + 2 y \sqrt{x + 3} + x + 3$

Step 5

Solve for $2 y \sqrt{x + 3}$ .

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Step 5.1

Rewrite the equation as $y^{2} + 2 y \sqrt{x + 3} + x + 3 = x - 3$ .

$y^{2} + 2 y \sqrt{x + 3} + x + 3 = x - 3$

Step 5.2

Move all terms not containing $2 y \sqrt{x + 3}$ to the right side of the equation.

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Step 5.2.1

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

$2 y \sqrt{x + 3} + x + 3 = x - 3 - y^{2}$

Step 5.2.2

Subtract $x$ from both sides of the equation.

$2 y \sqrt{x + 3} + 3 = x - 3 - y^{2} - x$

Step 5.2.3

Subtract $3$ from both sides of the equation.

$2 y \sqrt{x + 3} = x - 3 - y^{2} - x - 3$

Step 5.2.4

Combine the opposite terms in $x - 3 - y^{2} - x - 3$ .

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Step 5.2.4.1

Subtract $x$ from $x$ .

$2 y \sqrt{x + 3} = - 3 - y^{2} + 0 - 3$

Step 5.2.4.2

Add $- 3 - y^{2}$ and $0$ .

$2 y \sqrt{x + 3} = - 3 - y^{2} - 3$

Step 5.2.5

Subtract $3$ from $- 3$ .

$2 y \sqrt{x + 3} = - y^{2} - 6$

Step 6

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

${(2 y \sqrt{x + 3})}^{2} = {(- y^{2} - 6)}^{2}$

Step 7

Simplify each side of the equation.

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Step 7.1

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

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

Step 7.2

Simplify the left side.

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Step 7.2.1

Simplify ${(2 y {(x + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 7.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 7.2.1.1.1

Apply the product rule to $2 y {(x + 3)}^{\frac{1}{2}}$ .

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

Step 7.2.1.1.2

Apply the product rule to $2 y$ .

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

Step 7.2.1.2

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

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

Step 7.2.1.3

Multiply the exponents in ${({(x + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 7.2.1.3.1

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

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

Step 7.2.1.3.2

Cancel the common factor of $2$ .

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Step 7.2.1.3.2.1

Cancel the common factor.

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

Step 7.2.1.3.2.2

Rewrite the expression.

$4 y^{2} {(x + 3)}^{1} = {(- y^{2} - 6)}^{2}$

Step 7.2.1.4

Simplify.

$4 y^{2} (x + 3) = {(- y^{2} - 6)}^{2}$

Step 7.2.1.5

Apply the distributive property.

$4 y^{2} x + 4 y^{2} \cdot 3 = {(- y^{2} - 6)}^{2}$

Step 7.2.1.6

Multiply $3$ by $4$ .

$4 y^{2} x + 12 y^{2} = {(- y^{2} - 6)}^{2}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify ${(- y^{2} - 6)}^{2}$ .

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Step 7.3.1.1

Rewrite ${(- y^{2} - 6)}^{2}$ as $(- y^{2} - 6) (- y^{2} - 6)$ .

$4 y^{2} x + 12 y^{2} = (- y^{2} - 6) (- y^{2} - 6)$

Step 7.3.1.2

Expand $(- y^{2} - 6) (- y^{2} - 6)$ using the FOIL Method.

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Step 7.3.1.2.1

Apply the distributive property.

$4 y^{2} x + 12 y^{2} = - y^{2} (- y^{2} - 6) - 6 (- y^{2} - 6)$

Step 7.3.1.2.2

Apply the distributive property.

$4 y^{2} x + 12 y^{2} = - y^{2} (- y^{2}) - y^{2} \cdot - 6 - 6 (- y^{2} - 6)$

Step 7.3.1.2.3

Apply the distributive property.

$4 y^{2} x + 12 y^{2} = - y^{2} (- y^{2}) - y^{2} \cdot - 6 - 6 (- y^{2}) - 6 \cdot - 6$

Step 7.3.1.3

Simplify and combine like terms.

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Step 7.3.1.3.1

Simplify each term.

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Step 7.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$4 y^{2} x + 12 y^{2} = - 1 \cdot - 1 y^{2} y^{2} - y^{2} \cdot - 6 - 6 (- y^{2}) - 6 \cdot - 6$

Step 7.3.1.3.1.2

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 7.3.1.3.1.2.1

Move $y^{2}$ .

$4 y^{2} x + 12 y^{2} = - 1 \cdot - 1 (y^{2} y^{2}) - y^{2} \cdot - 6 - 6 (- y^{2}) - 6 \cdot - 6$

Step 7.3.1.3.1.2.2

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

$4 y^{2} x + 12 y^{2} = - 1 \cdot - 1 y^{2 + 2} - y^{2} \cdot - 6 - 6 (- y^{2}) - 6 \cdot - 6$

Step 7.3.1.3.1.2.3

Add $2$ and $2$ .

$4 y^{2} x + 12 y^{2} = - 1 \cdot - 1 y^{4} - y^{2} \cdot - 6 - 6 (- y^{2}) - 6 \cdot - 6$

Step 7.3.1.3.1.3

Multiply $- 1$ by $- 1$ .

$4 y^{2} x + 12 y^{2} = 1 y^{4} - y^{2} \cdot - 6 - 6 (- y^{2}) - 6 \cdot - 6$

Step 7.3.1.3.1.4

Multiply $y^{4}$ by $1$ .

$4 y^{2} x + 12 y^{2} = y^{4} - y^{2} \cdot - 6 - 6 (- y^{2}) - 6 \cdot - 6$

Step 7.3.1.3.1.5

Multiply $- 6$ by $- 1$ .

$4 y^{2} x + 12 y^{2} = y^{4} + 6 y^{2} - 6 (- y^{2}) - 6 \cdot - 6$

Step 7.3.1.3.1.6

Multiply $- 1$ by $- 6$ .

$4 y^{2} x + 12 y^{2} = y^{4} + 6 y^{2} + 6 y^{2} - 6 \cdot - 6$

Step 7.3.1.3.1.7

Multiply $- 6$ by $- 6$ .

$4 y^{2} x + 12 y^{2} = y^{4} + 6 y^{2} + 6 y^{2} + 36$

Step 7.3.1.3.2

Add $6 y^{2}$ and $6 y^{2}$ .

$4 y^{2} x + 12 y^{2} = y^{4} + 12 y^{2} + 36$

Step 8

Solve for $x$ .

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Step 8.1

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

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Step 8.1.1

Subtract $12 y^{2}$ from both sides of the equation.

$4 y^{2} x = y^{4} + 12 y^{2} + 36 - 12 y^{2}$

Step 8.1.2

Combine the opposite terms in $y^{4} + 12 y^{2} + 36 - 12 y^{2}$ .

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Step 8.1.2.1

Subtract $12 y^{2}$ from $12 y^{2}$ .

$4 y^{2} x = y^{4} + 0 + 36$

Step 8.1.2.2

Add $y^{4}$ and $0$ .

$4 y^{2} x = y^{4} + 36$

Step 8.2

Divide each term in $4 y^{2} x = y^{4} + 36$ by $4 y^{2}$ and simplify.

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Step 8.2.1

Divide each term in $4 y^{2} x = y^{4} + 36$ by $4 y^{2}$ .

$\frac{4 y^{2} x}{4 y^{2}} = \frac{y^{4}}{4 y^{2}} + \frac{36}{4 y^{2}}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $4$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{4 y^{2} x}{4 y^{2}} = \frac{y^{4}}{4 y^{2}} + \frac{36}{4 y^{2}}$

Step 8.2.2.1.2

Rewrite the expression.

$\frac{y^{2} x}{y^{2}} = \frac{y^{4}}{4 y^{2}} + \frac{36}{4 y^{2}}$

Step 8.2.2.2

Cancel the common factor of $y^{2}$ .

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Step 8.2.2.2.1

Cancel the common factor.

$\frac{y^{2} x}{y^{2}} = \frac{y^{4}}{4 y^{2}} + \frac{36}{4 y^{2}}$

Step 8.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{y^{4}}{4 y^{2}} + \frac{36}{4 y^{2}}$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Simplify each term.

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Step 8.2.3.1.1

Cancel the common factor of $y^{4}$ and $y^{2}$ .

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Step 8.2.3.1.1.1

Factor $y^{2}$ out of $y^{4}$ .

$x = \frac{y^{2} y^{2}}{4 y^{2}} + \frac{36}{4 y^{2}}$

Step 8.2.3.1.1.2

Cancel the common factors.

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Step 8.2.3.1.1.2.1

Factor $y^{2}$ out of $4 y^{2}$ .

$x = \frac{y^{2} y^{2}}{y^{2} \cdot 4} + \frac{36}{4 y^{2}}$

Step 8.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{y^{2} y^{2}}{y^{2} \cdot 4} + \frac{36}{4 y^{2}}$

Step 8.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{y^{2}}{4} + \frac{36}{4 y^{2}}$

Step 8.2.3.1.2

Cancel the common factor of $36$ and $4$ .

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Step 8.2.3.1.2.1

Factor $4$ out of $36$ .

$x = \frac{y^{2}}{4} + \frac{4 \cdot 9}{4 y^{2}}$

Step 8.2.3.1.2.2

Cancel the common factors.

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Step 8.2.3.1.2.2.1

Factor $4$ out of $4 y^{2}$ .

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

Step 8.2.3.1.2.2.2

Cancel the common factor.

$x = \frac{y^{2}}{4} + \frac{4 \cdot 9}{4 y^{2}}$

Step 8.2.3.1.2.2.3

Rewrite the expression.

$x = \frac{y^{2}}{4} + \frac{9}{y^{2}}$