Algebra Examples

$\sqrt{- 4 y + 1} + \sqrt{2 y + 4} = 3$

Step 1

Solve for $\sqrt{- 4 y + 1}$ .

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Step 1.1

Factor $2$ out of $2 y + 4$ .

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Step 1.1.1

Factor $2$ out of $2 y$ .

$\sqrt{- 4 y + 1} + \sqrt{2 (y) + 4} = 3$

Step 1.1.2

Factor $2$ out of $4$ .

$\sqrt{- 4 y + 1} + \sqrt{2 y + 2 \cdot 2} = 3$

Step 1.1.3

Factor $2$ out of $2 y + 2 \cdot 2$ .

$\sqrt{- 4 y + 1} + \sqrt{2 (y + 2)} = 3$

Step 1.2

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

$\sqrt{- 4 y + 1} = 3 - \sqrt{2 (y + 2)}$

Step 2

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

${\sqrt{- 4 y + 1}}^{2} = {(3 - \sqrt{2 (y + 2)})}^{2}$

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 $\sqrt{- 4 y + 1}$ as ${(- 4 y + 1)}^{\frac{1}{2}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(- 4 y + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(- 4 y + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

${(- 4 y + 1)}^{1} = {(3 - \sqrt{2 (y + 2)})}^{2}$

Step 3.2.1.2

Simplify.

$- 4 y + 1 = {(3 - \sqrt{2 (y + 2)})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

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

$- 4 y + 1 = (3 - \sqrt{2 (y + 2)}) (3 - \sqrt{2 (y + 2)})$

Step 3.3.1.2

Expand $(3 - \sqrt{2 (y + 2)}) (3 - \sqrt{2 (y + 2)})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$- 4 y + 1 = 3 (3 - \sqrt{2 (y + 2)}) - \sqrt{2 (y + 2)} (3 - \sqrt{2 (y + 2)})$

Step 3.3.1.2.2

Apply the distributive property.

$- 4 y + 1 = 3 \cdot 3 + 3 (- \sqrt{2 (y + 2)}) - \sqrt{2 (y + 2)} (3 - \sqrt{2 (y + 2)})$

Step 3.3.1.2.3

Apply the distributive property.

$- 4 y + 1 = 3 \cdot 3 + 3 (- \sqrt{2 (y + 2)}) - \sqrt{2 (y + 2)} \cdot 3 - \sqrt{2 (y + 2)} (- \sqrt{2 (y + 2)})$

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 $3$ by $3$ .

$- 4 y + 1 = 9 + 3 (- \sqrt{2 (y + 2)}) - \sqrt{2 (y + 2)} \cdot 3 - \sqrt{2 (y + 2)} (- \sqrt{2 (y + 2)})$

Step 3.3.1.3.1.2

Multiply $- 1$ by $3$ .

$- 4 y + 1 = 9 - 3 \sqrt{2 (y + 2)} - \sqrt{2 (y + 2)} \cdot 3 - \sqrt{2 (y + 2)} (- \sqrt{2 (y + 2)})$

Step 3.3.1.3.1.3

Multiply $3$ by $- 1$ .

$- 4 y + 1 = 9 - 3 \sqrt{2 (y + 2)} - 3 \sqrt{2 (y + 2)} - \sqrt{2 (y + 2)} (- \sqrt{2 (y + 2)})$

Step 3.3.1.3.1.4

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

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Step 3.3.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$- 4 y + 1 = 9 - 3 \sqrt{2 (y + 2)} - 3 \sqrt{2 (y + 2)} + 1 \sqrt{2 (y + 2)} \sqrt{2 (y + 2)}$

Step 3.3.1.3.1.4.2

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

$- 4 y + 1 = 9 - 3 \sqrt{2 (y + 2)} - 3 \sqrt{2 (y + 2)} + \sqrt{2 (y + 2)} \sqrt{2 (y + 2)}$

Step 3.3.1.3.1.4.3

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

$- 4 y + 1 = 9 - 3 \sqrt{2 (y + 2)} - 3 \sqrt{2 (y + 2)} + {\sqrt{2 (y + 2)}}^{1} \sqrt{2 (y + 2)}$

Step 3.3.1.3.1.4.4

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

$- 4 y + 1 = 9 - 3 \sqrt{2 (y + 2)} - 3 \sqrt{2 (y + 2)} + {\sqrt{2 (y + 2)}}^{1} {\sqrt{2 (y + 2)}}^{1}$

Step 3.3.1.3.1.4.5

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

$- 4 y + 1 = 9 - 3 \sqrt{2 (y + 2)} - 3 \sqrt{2 (y + 2)} + {\sqrt{2 (y + 2)}}^{1 + 1}$

Step 3.3.1.3.1.4.6

Add $1$ and $1$ .

$- 4 y + 1 = 9 - 3 \sqrt{2 (y + 2)} - 3 \sqrt{2 (y + 2)} + {\sqrt{2 (y + 2)}}^{2}$

Step 3.3.1.3.1.5

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

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Step 3.3.1.3.1.5.1

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

$- 4 y + 1 = 9 - 3 \sqrt{2 (y + 2)} - 3 \sqrt{2 (y + 2)} + {({(2 (y + 2))}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.5.2

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

$- 4 y + 1 = 9 - 3 \sqrt{2 (y + 2)} - 3 \sqrt{2 (y + 2)} + {(2 (y + 2))}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.5.3

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

$- 4 y + 1 = 9 - 3 \sqrt{2 (y + 2)} - 3 \sqrt{2 (y + 2)} + {(2 (y + 2))}^{\frac{2}{2}}$

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.

$- 4 y + 1 = 9 - 3 \sqrt{2 (y + 2)} - 3 \sqrt{2 (y + 2)} + {(2 (y + 2))}^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4.2

Rewrite the expression.

$- 4 y + 1 = 9 - 3 \sqrt{2 (y + 2)} - 3 \sqrt{2 (y + 2)} + {(2 (y + 2))}^{1}$

Step 3.3.1.3.1.5.5

Simplify.

$- 4 y + 1 = 9 - 3 \sqrt{2 (y + 2)} - 3 \sqrt{2 (y + 2)} + 2 (y + 2)$

Step 3.3.1.3.1.6

Apply the distributive property.

$- 4 y + 1 = 9 - 3 \sqrt{2 (y + 2)} - 3 \sqrt{2 (y + 2)} + 2 y + 2 \cdot 2$

Step 3.3.1.3.1.7

Multiply $2$ by $2$ .

$- 4 y + 1 = 9 - 3 \sqrt{2 (y + 2)} - 3 \sqrt{2 (y + 2)} + 2 y + 4$

Step 3.3.1.3.2

Add $9$ and $4$ .

$- 4 y + 1 = 13 - 3 \sqrt{2 (y + 2)} - 3 \sqrt{2 (y + 2)} + 2 y$

Step 3.3.1.3.3

Subtract $3 \sqrt{2 (y + 2)}$ from $- 3 \sqrt{2 (y + 2)}$ .

$- 4 y + 1 = 13 - 6 \sqrt{2 (y + 2)} + 2 y$

Step 4

Solve for $- 6 \sqrt{2 (y + 2)}$ .

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Step 4.1

Rewrite the equation as $13 - 6 \sqrt{2 (y + 2)} + 2 y = - 4 y + 1$ .

$13 - 6 \sqrt{2 (y + 2)} + 2 y = - 4 y + 1$

Step 4.2

Move all terms not containing $- 6 \sqrt{2 (y + 2)}$ to the right side of the equation.

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Step 4.2.1

Subtract $13$ from both sides of the equation.

$- 6 \sqrt{2 (y + 2)} + 2 y = - 4 y + 1 - 13$

Step 4.2.2

Subtract $2 y$ from both sides of the equation.

$- 6 \sqrt{2 (y + 2)} = - 4 y + 1 - 13 - 2 y$

Step 4.2.3

Subtract $2 y$ from $- 4 y$ .

$- 6 \sqrt{2 (y + 2)} = - 6 y + 1 - 13$

Step 4.2.4

Subtract $13$ from $1$ .

$- 6 \sqrt{2 (y + 2)} = - 6 y - 12$

Step 5

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

${(- 6 \sqrt{2 (y + 2)})}^{2} = {(- 6 y - 12)}^{2}$

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

${(- 6 {(2 (y + 2))}^{\frac{1}{2}})}^{2} = {(- 6 y - 12)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(- 6 {(2 (y + 2))}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the distributive property.

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

Step 6.2.1.2

Simplify the expression.

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Step 6.2.1.2.1

Multiply $2$ by $2$ .

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

Step 6.2.1.2.2

Apply the product rule to $- 6 {(2 y + 4)}^{\frac{1}{2}}$ .

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

Step 6.2.1.2.3

Raise $- 6$ to the power of $2$ .

$36 {({(2 y + 4)}^{\frac{1}{2}})}^{2} = {(- 6 y - 12)}^{2}$

Step 6.2.1.2.4

Multiply the exponents in ${({(2 y + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.2.4.1

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

$36 {(2 y + 4)}^{\frac{1}{2} \cdot 2} = {(- 6 y - 12)}^{2}$

Step 6.2.1.2.4.2

Cancel the common factor of $2$ .

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Step 6.2.1.2.4.2.1

Cancel the common factor.

$36 {(2 y + 4)}^{\frac{1}{2} \cdot 2} = {(- 6 y - 12)}^{2}$

Step 6.2.1.2.4.2.2

Rewrite the expression.

$36 {(2 y + 4)}^{1} = {(- 6 y - 12)}^{2}$

Step 6.2.1.3

Simplify.

$36 (2 y + 4) = {(- 6 y - 12)}^{2}$

Step 6.2.1.4

Apply the distributive property.

$36 (2 y) + 36 \cdot 4 = {(- 6 y - 12)}^{2}$

Step 6.2.1.5

Multiply.

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Step 6.2.1.5.1

Multiply $2$ by $36$ .

$72 y + 36 \cdot 4 = {(- 6 y - 12)}^{2}$

Step 6.2.1.5.2

Multiply $36$ by $4$ .

$72 y + 144 = {(- 6 y - 12)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify ${(- 6 y - 12)}^{2}$ .

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Step 6.3.1.1

Rewrite ${(- 6 y - 12)}^{2}$ as $(- 6 y - 12) (- 6 y - 12)$ .

$72 y + 144 = (- 6 y - 12) (- 6 y - 12)$

Step 6.3.1.2

Expand $(- 6 y - 12) (- 6 y - 12)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$72 y + 144 = - 6 y (- 6 y - 12) - 12 (- 6 y - 12)$

Step 6.3.1.2.2

Apply the distributive property.

$72 y + 144 = - 6 y (- 6 y) - 6 y \cdot - 12 - 12 (- 6 y - 12)$

Step 6.3.1.2.3

Apply the distributive property.

$72 y + 144 = - 6 y (- 6 y) - 6 y \cdot - 12 - 12 (- 6 y) - 12 \cdot - 12$

Step 6.3.1.3

Simplify and combine like terms.

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Step 6.3.1.3.1

Simplify each term.

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Step 6.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$72 y + 144 = - 6 \cdot - 6 y \cdot y - 6 y \cdot - 12 - 12 (- 6 y) - 12 \cdot - 12$

Step 6.3.1.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 6.3.1.3.1.2.1

Move $y$ .

$72 y + 144 = - 6 \cdot - 6 (y \cdot y) - 6 y \cdot - 12 - 12 (- 6 y) - 12 \cdot - 12$

Step 6.3.1.3.1.2.2

Multiply $y$ by $y$ .

$72 y + 144 = - 6 \cdot - 6 y^{2} - 6 y \cdot - 12 - 12 (- 6 y) - 12 \cdot - 12$

Step 6.3.1.3.1.3

Multiply $- 6$ by $- 6$ .

$72 y + 144 = 36 y^{2} - 6 y \cdot - 12 - 12 (- 6 y) - 12 \cdot - 12$

Step 6.3.1.3.1.4

Multiply $- 12$ by $- 6$ .

$72 y + 144 = 36 y^{2} + 72 y - 12 (- 6 y) - 12 \cdot - 12$

Step 6.3.1.3.1.5

Multiply $- 6$ by $- 12$ .

$72 y + 144 = 36 y^{2} + 72 y + 72 y - 12 \cdot - 12$

Step 6.3.1.3.1.6

Multiply $- 12$ by $- 12$ .

$72 y + 144 = 36 y^{2} + 72 y + 72 y + 144$

Step 6.3.1.3.2

Add $72 y$ and $72 y$ .

$72 y + 144 = 36 y^{2} + 144 y + 144$

Step 7

Solve for $y$ .

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Step 7.1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$36 y^{2} + 144 y + 144 = 72 y + 144$

Step 7.2

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

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Step 7.2.1

Subtract $72 y$ from both sides of the equation.

$36 y^{2} + 144 y + 144 - 72 y = 144$

Step 7.2.2

Subtract $72 y$ from $144 y$ .

$36 y^{2} + 72 y + 144 = 144$

Step 7.3

Subtract $144$ from both sides of the equation.

$36 y^{2} + 72 y + 144 - 144 = 0$

Step 7.4

Combine the opposite terms in $36 y^{2} + 72 y + 144 - 144$ .

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Step 7.4.1

Subtract $144$ from $144$ .

$36 y^{2} + 72 y + 0 = 0$

Step 7.4.2

Add $36 y^{2} + 72 y$ and $0$ .

$36 y^{2} + 72 y = 0$

Step 7.5

Factor $36 y$ out of $36 y^{2} + 72 y$ .

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Step 7.5.1

Factor $36 y$ out of $36 y^{2}$ .

$36 y (y) + 72 y = 0$

Step 7.5.2

Factor $36 y$ out of $72 y$ .

$36 y (y) + 36 y (2) = 0$

Step 7.5.3

Factor $36 y$ out of $36 y (y) + 36 y (2)$ .

$36 y (y + 2) = 0$

Step 7.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y = 0$

$y + 2 = 0$

Step 7.7

Set $y$ equal to $0$ .

$y = 0$

Step 7.8

Set $y + 2$ equal to $0$ and solve for $y$ .

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Step 7.8.1

Set $y + 2$ equal to $0$ .

$y + 2 = 0$

Step 7.8.2

Subtract $2$ from both sides of the equation.

$y = - 2$

Step 7.9

The final solution is all the values that make $36 y (y + 2) = 0$ true.

$y = 0, - 2$