Algebra Examples

$\sqrt{y} + \sqrt{y + 2} = 6$

Step 1

Subtract $\sqrt{y + 2}$ from both sides of the equation.

$\sqrt{y} = 6 - \sqrt{y + 2}$

Step 2

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

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

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 $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{1} = {(6 - \sqrt{y + 2})}^{2}$

Step 3.2.1.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

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

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

Step 3.3.1.2

Expand $(6 - \sqrt{y + 2}) (6 - \sqrt{y + 2})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

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

Step 3.3.1.2.2

Apply the distributive property.

$y = 6 \cdot 6 + 6 (- \sqrt{y + 2}) - \sqrt{y + 2} (6 - \sqrt{y + 2})$

Step 3.3.1.2.3

Apply the distributive property.

$y = 6 \cdot 6 + 6 (- \sqrt{y + 2}) - \sqrt{y + 2} \cdot 6 - \sqrt{y + 2} (- \sqrt{y + 2})$

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

Multiply $6$ by $6$ .

$y = 36 + 6 (- \sqrt{y + 2}) - \sqrt{y + 2} \cdot 6 - \sqrt{y + 2} (- \sqrt{y + 2})$

Step 3.3.1.3.1.2

Multiply $- 1$ by $6$ .

$y = 36 - 6 \sqrt{y + 2} - \sqrt{y + 2} \cdot 6 - \sqrt{y + 2} (- \sqrt{y + 2})$

Step 3.3.1.3.1.3

Multiply $6$ by $- 1$ .

$y = 36 - 6 \sqrt{y + 2} - 6 \sqrt{y + 2} - \sqrt{y + 2} (- \sqrt{y + 2})$

Step 3.3.1.3.1.4

Multiply $- \sqrt{y + 2} (- \sqrt{y + 2})$ .

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Step 3.3.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$y = 36 - 6 \sqrt{y + 2} - 6 \sqrt{y + 2} + 1 \sqrt{y + 2} \sqrt{y + 2}$

Step 3.3.1.3.1.4.2

Multiply $\sqrt{y + 2}$ by $1$ .

$y = 36 - 6 \sqrt{y + 2} - 6 \sqrt{y + 2} + \sqrt{y + 2} \sqrt{y + 2}$

Step 3.3.1.3.1.4.3

Raise $\sqrt{y + 2}$ to the power of $1$ .

$y = 36 - 6 \sqrt{y + 2} - 6 \sqrt{y + 2} + {\sqrt{y + 2}}^{1} \sqrt{y + 2}$

Step 3.3.1.3.1.4.4

Raise $\sqrt{y + 2}$ to the power of $1$ .

$y = 36 - 6 \sqrt{y + 2} - 6 \sqrt{y + 2} + {\sqrt{y + 2}}^{1} {\sqrt{y + 2}}^{1}$

Step 3.3.1.3.1.4.5

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

$y = 36 - 6 \sqrt{y + 2} - 6 \sqrt{y + 2} + {\sqrt{y + 2}}^{1 + 1}$

Step 3.3.1.3.1.4.6

Add $1$ and $1$ .

$y = 36 - 6 \sqrt{y + 2} - 6 \sqrt{y + 2} + {\sqrt{y + 2}}^{2}$

Step 3.3.1.3.1.5

Rewrite ${\sqrt{y + 2}}^{2}$ as $y + 2$ .

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Step 3.3.1.3.1.5.1

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

$y = 36 - 6 \sqrt{y + 2} - 6 \sqrt{y + 2} + {({(y + 2)}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.5.2

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

$y = 36 - 6 \sqrt{y + 2} - 6 \sqrt{y + 2} + {(y + 2)}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.5.3

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

$y = 36 - 6 \sqrt{y + 2} - 6 \sqrt{y + 2} + {(y + 2)}^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.5.4.1

Cancel the common factor.

$y = 36 - 6 \sqrt{y + 2} - 6 \sqrt{y + 2} + {(y + 2)}^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4.2

Rewrite the expression.

$y = 36 - 6 \sqrt{y + 2} - 6 \sqrt{y + 2} + {(y + 2)}^{1}$

Step 3.3.1.3.1.5.5

Simplify.

$y = 36 - 6 \sqrt{y + 2} - 6 \sqrt{y + 2} + y + 2$

Step 3.3.1.3.2

Add $36$ and $2$ .

$y = 38 - 6 \sqrt{y + 2} - 6 \sqrt{y + 2} + y$

Step 3.3.1.3.3

Subtract $6 \sqrt{y + 2}$ from $- 6 \sqrt{y + 2}$ .

$y = 38 - 12 \sqrt{y + 2} + y$

Step 4

Solve for $- 12 \sqrt{y + 2}$ .

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Step 4.1

Rewrite the equation as $38 - 12 \sqrt{y + 2} + y = y$ .

$38 - 12 \sqrt{y + 2} + y = y$

Step 4.2

Move all terms not containing $- 12 \sqrt{y + 2}$ to the right side of the equation.

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Step 4.2.1

Subtract $38$ from both sides of the equation.

$- 12 \sqrt{y + 2} + y = y - 38$

Step 4.2.2

Subtract $y$ from both sides of the equation.

$- 12 \sqrt{y + 2} = y - 38 - y$

Step 4.2.3

Combine the opposite terms in $y - 38 - y$ .

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Step 4.2.3.1

Subtract $y$ from $y$ .

$- 12 \sqrt{y + 2} = 0 - 38$

Step 4.2.3.2

Subtract $38$ from $0$ .

$- 12 \sqrt{y + 2} = - 38$

Step 5

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

${(- 12 \sqrt{y + 2})}^{2} = {(- 38)}^{2}$

Step 6

Simplify each side of the equation.

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Step 6.1

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

${(- 12 {(y + 2)}^{\frac{1}{2}})}^{2} = {(- 38)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(- 12 {(y + 2)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $- 12 {(y + 2)}^{\frac{1}{2}}$ .

${(- 12)}^{2} {({(y + 2)}^{\frac{1}{2}})}^{2} = {(- 38)}^{2}$

Step 6.2.1.2

Raise $- 12$ to the power of $2$ .

$144 {({(y + 2)}^{\frac{1}{2}})}^{2} = {(- 38)}^{2}$

Step 6.2.1.3

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

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Step 6.2.1.3.1

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

$144 {(y + 2)}^{\frac{1}{2} \cdot 2} = {(- 38)}^{2}$

Step 6.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.2.1.3.2.1

Cancel the common factor.

$144 {(y + 2)}^{\frac{1}{2} \cdot 2} = {(- 38)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$144 {(y + 2)}^{1} = {(- 38)}^{2}$

Step 6.2.1.4

Simplify.

$144 (y + 2) = {(- 38)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$144 y + 144 \cdot 2 = {(- 38)}^{2}$

Step 6.2.1.6

Multiply $144$ by $2$ .

$144 y + 288 = {(- 38)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Raise $- 38$ to the power of $2$ .

$144 y + 288 = 1444$

Step 7

Solve for $y$ .

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Step 7.1

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

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Step 7.1.1

Subtract $288$ from both sides of the equation.

$144 y = 1444 - 288$

Step 7.1.2

Subtract $288$ from $1444$ .

$144 y = 1156$

Step 7.2

Divide each term in $144 y = 1156$ by $144$ and simplify.

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Step 7.2.1

Divide each term in $144 y = 1156$ by $144$ .

$\frac{144 y}{144} = \frac{1156}{144}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $144$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{144 y}{144} = \frac{1156}{144}$

Step 7.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{1156}{144}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Cancel the common factor of $1156$ and $144$ .

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Step 7.2.3.1.1

Factor $4$ out of $1156$ .

$y = \frac{4 (289)}{144}$

Step 7.2.3.1.2

Cancel the common factors.

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Step 7.2.3.1.2.1

Factor $4$ out of $144$ .

$y = \frac{4 \cdot 289}{4 \cdot 36}$

Step 7.2.3.1.2.2

Cancel the common factor.

$y = \frac{4 \cdot 289}{4 \cdot 36}$

Step 7.2.3.1.2.3

Rewrite the expression.

$y = \frac{289}{36}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$y = \frac{289}{36}$

Decimal Form:

$y = 8.02\overline{7}$

Mixed Number Form:

$y = 8 \frac{1}{36}$