Algebra Examples

$\sqrt{4 y + 20} - \sqrt{y - 4} = 6$

Step 1

Add $\sqrt{y - 4}$ to both sides of the equation.

$\sqrt{4 y + 20} = 6 + \sqrt{y - 4}$

Step 2

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

${\sqrt{4 y + 20}}^{2} = {(6 + \sqrt{y - 4})}^{2}$

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 $\sqrt{4 y + 20}$ as ${(4 y + 20)}^{\frac{1}{2}}$ .

${({(4 y + 20)}^{\frac{1}{2}})}^{2} = {(6 + \sqrt{y - 4})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(4 y + 20)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(4 y + 20)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(4 y + 20)}^{\frac{1}{2} \cdot 2} = {(6 + \sqrt{y - 4})}^{2}$

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.

${(4 y + 20)}^{\frac{1}{2} \cdot 2} = {(6 + \sqrt{y - 4})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(4 y + 20)}^{1} = {(6 + \sqrt{y - 4})}^{2}$

Step 3.2.1.2

Simplify.

$4 y + 20 = {(6 + \sqrt{y - 4})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(6 + \sqrt{y - 4})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(6 + \sqrt{y - 4})}^{2}$ as $(6 + \sqrt{y - 4}) (6 + \sqrt{y - 4})$ .

$4 y + 20 = (6 + \sqrt{y - 4}) (6 + \sqrt{y - 4})$

Step 3.3.1.2

Expand $(6 + \sqrt{y - 4}) (6 + \sqrt{y - 4})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$4 y + 20 = 6 (6 + \sqrt{y - 4}) + \sqrt{y - 4} (6 + \sqrt{y - 4})$

Step 3.3.1.2.2

Apply the distributive property.

$4 y + 20 = 6 \cdot 6 + 6 \sqrt{y - 4} + \sqrt{y - 4} (6 + \sqrt{y - 4})$

Step 3.3.1.2.3

Apply the distributive property.

$4 y + 20 = 6 \cdot 6 + 6 \sqrt{y - 4} + \sqrt{y - 4} \cdot 6 + \sqrt{y - 4} \sqrt{y - 4}$

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 $6$ .

$4 y + 20 = 36 + 6 \sqrt{y - 4} + \sqrt{y - 4} \cdot 6 + \sqrt{y - 4} \sqrt{y - 4}$

Step 3.3.1.3.1.2

Move $6$ to the left of $\sqrt{y - 4}$ .

$4 y + 20 = 36 + 6 \sqrt{y - 4} + 6 \cdot \sqrt{y - 4} + \sqrt{y - 4} \sqrt{y - 4}$

Step 3.3.1.3.1.3

Multiply $\sqrt{y - 4} \sqrt{y - 4}$ .

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Step 3.3.1.3.1.3.1

Raise $\sqrt{y - 4}$ to the power of $1$ .

$4 y + 20 = 36 + 6 \sqrt{y - 4} + 6 \sqrt{y - 4} + {\sqrt{y - 4}}^{1} \sqrt{y - 4}$

Step 3.3.1.3.1.3.2

Raise $\sqrt{y - 4}$ to the power of $1$ .

$4 y + 20 = 36 + 6 \sqrt{y - 4} + 6 \sqrt{y - 4} + {\sqrt{y - 4}}^{1} {\sqrt{y - 4}}^{1}$

Step 3.3.1.3.1.3.3

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

$4 y + 20 = 36 + 6 \sqrt{y - 4} + 6 \sqrt{y - 4} + {\sqrt{y - 4}}^{1 + 1}$

Step 3.3.1.3.1.3.4

Add $1$ and $1$ .

$4 y + 20 = 36 + 6 \sqrt{y - 4} + 6 \sqrt{y - 4} + {\sqrt{y - 4}}^{2}$

Step 3.3.1.3.1.4

Rewrite ${\sqrt{y - 4}}^{2}$ as $y - 4$ .

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Step 3.3.1.3.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y - 4}$ as ${(y - 4)}^{\frac{1}{2}}$ .

$4 y + 20 = 36 + 6 \sqrt{y - 4} + 6 \sqrt{y - 4} + {({(y - 4)}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.4.2

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

$4 y + 20 = 36 + 6 \sqrt{y - 4} + 6 \sqrt{y - 4} + {(y - 4)}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.4.3

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

$4 y + 20 = 36 + 6 \sqrt{y - 4} + 6 \sqrt{y - 4} + {(y - 4)}^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.4.4.1

Cancel the common factor.

$4 y + 20 = 36 + 6 \sqrt{y - 4} + 6 \sqrt{y - 4} + {(y - 4)}^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4.2

Rewrite the expression.

$4 y + 20 = 36 + 6 \sqrt{y - 4} + 6 \sqrt{y - 4} + {(y - 4)}^{1}$

Step 3.3.1.3.1.4.5

Simplify.

$4 y + 20 = 36 + 6 \sqrt{y - 4} + 6 \sqrt{y - 4} + y - 4$

Step 3.3.1.3.2

Subtract $4$ from $36$ .

$4 y + 20 = 32 + 6 \sqrt{y - 4} + 6 \sqrt{y - 4} + y$

Step 3.3.1.3.3

Add $6 \sqrt{y - 4}$ and $6 \sqrt{y - 4}$ .

$4 y + 20 = 32 + 12 \sqrt{y - 4} + y$

Step 4

Solve for $12 \sqrt{y - 4}$ .

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Step 4.1

Rewrite the equation as $32 + 12 \sqrt{y - 4} + y = 4 y + 20$ .

$32 + 12 \sqrt{y - 4} + y = 4 y + 20$

Step 4.2

Move all terms not containing $12 \sqrt{y - 4}$ to the right side of the equation.

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Step 4.2.1

Subtract $32$ from both sides of the equation.

$12 \sqrt{y - 4} + y = 4 y + 20 - 32$

Step 4.2.2

Subtract $y$ from both sides of the equation.

$12 \sqrt{y - 4} = 4 y + 20 - 32 - y$

Step 4.2.3

Subtract $y$ from $4 y$ .

$12 \sqrt{y - 4} = 3 y + 20 - 32$

Step 4.2.4

Subtract $32$ from $20$ .

$12 \sqrt{y - 4} = 3 y - 12$

Step 5

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

${(12 \sqrt{y - 4})}^{2} = {(3 y - 12)}^{2}$

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

${(12 {(y - 4)}^{\frac{1}{2}})}^{2} = {(3 y - 12)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(12 {(y - 4)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $12 {(y - 4)}^{\frac{1}{2}}$ .

$12^{2} {({(y - 4)}^{\frac{1}{2}})}^{2} = {(3 y - 12)}^{2}$

Step 6.2.1.2

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

$144 {({(y - 4)}^{\frac{1}{2}})}^{2} = {(3 y - 12)}^{2}$

Step 6.2.1.3

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

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Step 6.2.1.3.1

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

$144 {(y - 4)}^{\frac{1}{2} \cdot 2} = {(3 y - 12)}^{2}$

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.

$144 {(y - 4)}^{\frac{1}{2} \cdot 2} = {(3 y - 12)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$144 {(y - 4)}^{1} = {(3 y - 12)}^{2}$

Step 6.2.1.4

Simplify.

$144 (y - 4) = {(3 y - 12)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$144 y + 144 \cdot - 4 = {(3 y - 12)}^{2}$

Step 6.2.1.6

Multiply $144$ by $- 4$ .

$144 y - 576 = {(3 y - 12)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify ${(3 y - 12)}^{2}$ .

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Step 6.3.1.1

Rewrite ${(3 y - 12)}^{2}$ as $(3 y - 12) (3 y - 12)$ .

$144 y - 576 = (3 y - 12) (3 y - 12)$

Step 6.3.1.2

Expand $(3 y - 12) (3 y - 12)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$144 y - 576 = 3 y (3 y - 12) - 12 (3 y - 12)$

Step 6.3.1.2.2

Apply the distributive property.

$144 y - 576 = 3 y (3 y) + 3 y \cdot - 12 - 12 (3 y - 12)$

Step 6.3.1.2.3

Apply the distributive property.

$144 y - 576 = 3 y (3 y) + 3 y \cdot - 12 - 12 (3 y) - 12 \cdot - 12$

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.

$144 y - 576 = 3 \cdot 3 y \cdot y + 3 y \cdot - 12 - 12 (3 y) - 12 \cdot - 12$

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 $y$ .

$144 y - 576 = 3 \cdot 3 (y \cdot y) + 3 y \cdot - 12 - 12 (3 y) - 12 \cdot - 12$

Step 6.3.1.3.1.2.2

Multiply $y$ by $y$ .

$144 y - 576 = 3 \cdot 3 y^{2} + 3 y \cdot - 12 - 12 (3 y) - 12 \cdot - 12$

Step 6.3.1.3.1.3

Multiply $3$ by $3$ .

$144 y - 576 = 9 y^{2} + 3 y \cdot - 12 - 12 (3 y) - 12 \cdot - 12$

Step 6.3.1.3.1.4

Multiply $- 12$ by $3$ .

$144 y - 576 = 9 y^{2} - 36 y - 12 (3 y) - 12 \cdot - 12$

Step 6.3.1.3.1.5

Multiply $3$ by $- 12$ .

$144 y - 576 = 9 y^{2} - 36 y - 36 y - 12 \cdot - 12$

Step 6.3.1.3.1.6

Multiply $- 12$ by $- 12$ .

$144 y - 576 = 9 y^{2} - 36 y - 36 y + 144$

Step 6.3.1.3.2

Subtract $36 y$ from $- 36 y$ .

$144 y - 576 = 9 y^{2} - 72 y + 144$

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.

$9 y^{2} - 72 y + 144 = 144 y - 576$

Step 7.2

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

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Step 7.2.1

Subtract $144 y$ from both sides of the equation.

$9 y^{2} - 72 y + 144 - 144 y = - 576$

Step 7.2.2

Subtract $144 y$ from $- 72 y$ .

$9 y^{2} - 216 y + 144 = - 576$

Step 7.3

Add $576$ to both sides of the equation.

$9 y^{2} - 216 y + 144 + 576 = 0$

Step 7.4

Add $144$ and $576$ .

$9 y^{2} - 216 y + 720 = 0$

Step 7.5

Factor the left side of the equation.

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Step 7.5.1

Factor $9$ out of $9 y^{2} - 216 y + 720$ .

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Step 7.5.1.1

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

$9 (y^{2}) - 216 y + 720 = 0$

Step 7.5.1.2

Factor $9$ out of $- 216 y$ .

$9 (y^{2}) + 9 (- 24 y) + 720 = 0$

Step 7.5.1.3

Factor $9$ out of $720$ .

$9 y^{2} + 9 (- 24 y) + 9 \cdot 80 = 0$

Step 7.5.1.4

Factor $9$ out of $9 y^{2} + 9 (- 24 y)$ .

$9 (y^{2} - 24 y) + 9 \cdot 80 = 0$

Step 7.5.1.5

Factor $9$ out of $9 (y^{2} - 24 y) + 9 \cdot 80$ .

$9 (y^{2} - 24 y + 80) = 0$

Step 7.5.2

Factor.

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Step 7.5.2.1

Factor $y^{2} - 24 y + 80$ using the AC method.

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Step 7.5.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $80$ and whose sum is $- 24$ .

$- 20, - 4$

Step 7.5.2.1.2

Write the factored form using these integers.

$9 ((y - 20) (y - 4)) = 0$

Step 7.5.2.2

Remove unnecessary parentheses.

$9 (y - 20) (y - 4) = 0$

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 $0$ .

$y - 20 = 0$

$y - 4 = 0$

Step 7.7

Set $y - 20$ equal to $0$ and solve for $y$ .

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Step 7.7.1

Set $y - 20$ equal to $0$ .

$y - 20 = 0$

Step 7.7.2

Add $20$ to both sides of the equation.

$y = 20$

Step 7.8

Set $y - 4$ equal to $0$ and solve for $y$ .

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Step 7.8.1

Set $y - 4$ equal to $0$ .

$y - 4 = 0$

Step 7.8.2

Add $4$ to both sides of the equation.

$y = 4$

Step 7.9

The final solution is all the values that make $9 (y - 20) (y - 4) = 0$ true.

$y = 20, 4$