Precalculus Examples

$\sqrt{5 y + 4} + 2 = y$

Step 1

Subtract $2$ from both sides of the equation.

$\sqrt{5 y + 4} = y - 2$

Step 2

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

${\sqrt{5 y + 4}}^{2} = {(y - 2)}^{2}$

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 $\sqrt{5 y + 4}$ as ${(5 y + 4)}^{\frac{1}{2}}$ .

${({(5 y + 4)}^{\frac{1}{2}})}^{2} = {(y - 2)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(5 y + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(5 y + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(5 y + 4)}^{\frac{1}{2} \cdot 2} = {(y - 2)}^{2}$

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.

${(5 y + 4)}^{\frac{1}{2} \cdot 2} = {(y - 2)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(5 y + 4)}^{1} = {(y - 2)}^{2}$

Step 3.2.1.2

Simplify.

$5 y + 4 = {(y - 2)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

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

$5 y + 4 = (y - 2) (y - 2)$

Step 3.3.1.2

Expand $(y - 2) (y - 2)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$5 y + 4 = y (y - 2) - 2 (y - 2)$

Step 3.3.1.2.2

Apply the distributive property.

$5 y + 4 = y \cdot y + y \cdot - 2 - 2 (y - 2)$

Step 3.3.1.2.3

Apply the distributive property.

$5 y + 4 = y \cdot y + y \cdot - 2 - 2 y - 2 \cdot - 2$

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 $y$ by $y$ .

$5 y + 4 = y^{2} + y \cdot - 2 - 2 y - 2 \cdot - 2$

Step 3.3.1.3.1.2

Move $- 2$ to the left of $y$ .

$5 y + 4 = y^{2} - 2 \cdot y - 2 y - 2 \cdot - 2$

Step 3.3.1.3.1.3

Multiply $- 2$ by $- 2$ .

$5 y + 4 = y^{2} - 2 y - 2 y + 4$

Step 3.3.1.3.2

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

$5 y + 4 = y^{2} - 4 y + 4$

Step 4

Solve for $y$ .

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Step 4.1

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

$y^{2} - 4 y + 4 = 5 y + 4$

Step 4.2

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

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Step 4.2.1

Subtract $5 y$ from both sides of the equation.

$y^{2} - 4 y + 4 - 5 y = 4$

Step 4.2.2

Subtract $5 y$ from $- 4 y$ .

$y^{2} - 9 y + 4 = 4$

Step 4.3

Subtract $4$ from both sides of the equation.

$y^{2} - 9 y + 4 - 4 = 0$

Step 4.4

Combine the opposite terms in $y^{2} - 9 y + 4 - 4$ .

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Step 4.4.1

Subtract $4$ from $4$ .

$y^{2} - 9 y + 0 = 0$

Step 4.4.2

Add $y^{2} - 9 y$ and $0$ .

$y^{2} - 9 y = 0$

Step 4.5

Factor $y$ out of $y^{2} - 9 y$ .

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Step 4.5.1

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

$y \cdot y - 9 y = 0$

Step 4.5.2

Factor $y$ out of $- 9 y$ .

$y \cdot y + y \cdot - 9 = 0$

Step 4.5.3

Factor $y$ out of $y \cdot y + y \cdot - 9$ .

$y (y - 9) = 0$

Step 4.6

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

$y = 0$

$y - 9 = 0$

Step 4.7

Set $y$ equal to $0$ .

$y = 0$

Step 4.8

Set $y - 9$ equal to $0$ and solve for $y$ .

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Step 4.8.1

Set $y - 9$ equal to $0$ .

$y - 9 = 0$

Step 4.8.2

Add $9$ to both sides of the equation.

$y = 9$

Step 4.9

The final solution is all the values that make $y (y - 9) = 0$ true.

$y = 0, 9$

Step 5

Exclude the solutions that do not make $\sqrt{5 y + 4} + 2 = y$ true.

$y = 9$