Solve for y square root of 4y+5- square root of y-1=3

$\sqrt{4 y + 5} - \sqrt{y - 1} = 3$

Step 1

Add $\sqrt{y - 1}$ to both sides of the equation.

$\sqrt{4 y + 5} = 3 + \sqrt{y - 1}$

Step 2

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

${\sqrt{4 y + 5}}^{2} = {(3 + \sqrt{y - 1})}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 y + 5}$ as ${(4 y + 5)}^{\frac{1}{2}}$ .

${({(4 y + 5)}^{\frac{1}{2}})}^{2} = {(3 + \sqrt{y - 1})}^{2}$

Simplify the left side.

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

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

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

${(4 y + 5)}^{\frac{1}{2} \cdot 2} = {(3 + \sqrt{y - 1})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 y + 5)}^{\frac{1}{2} \cdot 2} = {(3 + \sqrt{y - 1})}^{2}$

Rewrite the expression.

${(4 y + 5)}^{1} = {(3 + \sqrt{y - 1})}^{2}$

Simplify.

$4 y + 5 = {(3 + \sqrt{y - 1})}^{2}$

Simplify the right side.

Simplify ${(3 + \sqrt{y - 1})}^{2}$ .

Rewrite ${(3 + \sqrt{y - 1})}^{2}$ as $(3 + \sqrt{y - 1}) (3 + \sqrt{y - 1})$ .

$4 y + 5 = (3 + \sqrt{y - 1}) (3 + \sqrt{y - 1})$

Expand $(3 + \sqrt{y - 1}) (3 + \sqrt{y - 1})$ using the FOIL Method.

Apply the distributive property.

$4 y + 5 = 3 (3 + \sqrt{y - 1}) + \sqrt{y - 1} (3 + \sqrt{y - 1})$

Apply the distributive property.

$4 y + 5 = 3 \cdot 3 + 3 \sqrt{y - 1} + \sqrt{y - 1} (3 + \sqrt{y - 1})$

Apply the distributive property.

$4 y + 5 = 3 \cdot 3 + 3 \sqrt{y - 1} + \sqrt{y - 1} \cdot 3 + \sqrt{y - 1} \sqrt{y - 1}$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$4 y + 5 = 9 + 3 \sqrt{y - 1} + \sqrt{y - 1} \cdot 3 + \sqrt{y - 1} \sqrt{y - 1}$

Move $3$ to the left of $\sqrt{y - 1}$ .

$4 y + 5 = 9 + 3 \sqrt{y - 1} + 3 \cdot \sqrt{y - 1} + \sqrt{y - 1} \sqrt{y - 1}$

Multiply $\sqrt{y - 1} \sqrt{y - 1}$ .

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

$4 y + 5 = 9 + 3 \sqrt{y - 1} + 3 \sqrt{y - 1} + {\sqrt{y - 1}}^{1} \sqrt{y - 1}$

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

$4 y + 5 = 9 + 3 \sqrt{y - 1} + 3 \sqrt{y - 1} + {\sqrt{y - 1}}^{1} {\sqrt{y - 1}}^{1}$

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

$4 y + 5 = 9 + 3 \sqrt{y - 1} + 3 \sqrt{y - 1} + {\sqrt{y - 1}}^{1 + 1}$

Add $1$ and $1$ .

$4 y + 5 = 9 + 3 \sqrt{y - 1} + 3 \sqrt{y - 1} + {\sqrt{y - 1}}^{2}$

Rewrite ${\sqrt{y - 1}}^{2}$ as $y - 1$ .

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

$4 y + 5 = 9 + 3 \sqrt{y - 1} + 3 \sqrt{y - 1} + {({(y - 1)}^{\frac{1}{2}})}^{2}$

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

$4 y + 5 = 9 + 3 \sqrt{y - 1} + 3 \sqrt{y - 1} + {(y - 1)}^{\frac{1}{2} \cdot 2}$

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

$4 y + 5 = 9 + 3 \sqrt{y - 1} + 3 \sqrt{y - 1} + {(y - 1)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 y + 5 = 9 + 3 \sqrt{y - 1} + 3 \sqrt{y - 1} + {(y - 1)}^{\frac{2}{2}}$

Rewrite the expression.

$4 y + 5 = 9 + 3 \sqrt{y - 1} + 3 \sqrt{y - 1} + {(y - 1)}^{1}$

Simplify.

$4 y + 5 = 9 + 3 \sqrt{y - 1} + 3 \sqrt{y - 1} + y - 1$

Subtract $1$ from $9$ .

$4 y + 5 = 8 + 3 \sqrt{y - 1} + 3 \sqrt{y - 1} + y$

Add $3 \sqrt{y - 1}$ and $3 \sqrt{y - 1}$ .

$4 y + 5 = 8 + 6 \sqrt{y - 1} + y$

Step 4

Solve for $6 \sqrt{y - 1}$ .

Rewrite the equation as $8 + 6 \sqrt{y - 1} + y = 4 y + 5$ .

$8 + 6 \sqrt{y - 1} + y = 4 y + 5$

Move all terms not containing $6 \sqrt{y - 1}$ to the right side of the equation.

Subtract $8$ from both sides of the equation.

$6 \sqrt{y - 1} + y = 4 y + 5 - 8$

Subtract $y$ from both sides of the equation.

$6 \sqrt{y - 1} = 4 y + 5 - 8 - y$

Subtract $y$ from $4 y$ .

$6 \sqrt{y - 1} = 3 y + 5 - 8$

Subtract $8$ from $5$ .

$6 \sqrt{y - 1} = 3 y - 3$

Step 5

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

${(6 \sqrt{y - 1})}^{2} = {(3 y - 3)}^{2}$

Step 6

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

$36 {({(y - 1)}^{\frac{1}{2}})}^{2} = {(3 y - 3)}^{2}$

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

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

$36 {(y - 1)}^{\frac{1}{2} \cdot 2} = {(3 y - 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$36 {(y - 1)}^{\frac{1}{2} \cdot 2} = {(3 y - 3)}^{2}$

Rewrite the expression.

$36 {(y - 1)}^{1} = {(3 y - 3)}^{2}$

Simplify.

$36 (y - 1) = {(3 y - 3)}^{2}$

Apply the distributive property.

$36 y + 36 \cdot - 1 = {(3 y - 3)}^{2}$

Multiply $36$ by $- 1$ .

$36 y - 36 = {(3 y - 3)}^{2}$

Simplify the right side.

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

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

$36 y - 36 = (3 y - 3) (3 y - 3)$

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

Apply the distributive property.

$36 y - 36 = 3 y (3 y - 3) - 3 (3 y - 3)$

Apply the distributive property.

$36 y - 36 = 3 y (3 y) + 3 y \cdot - 3 - 3 (3 y - 3)$

Apply the distributive property.

$36 y - 36 = 3 y (3 y) + 3 y \cdot - 3 - 3 (3 y) - 3 \cdot - 3$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$36 y - 36 = 3 \cdot 3 y \cdot y + 3 y \cdot - 3 - 3 (3 y) - 3 \cdot - 3$

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

Move $y$ .

$36 y - 36 = 3 \cdot 3 (y \cdot y) + 3 y \cdot - 3 - 3 (3 y) - 3 \cdot - 3$

Multiply $y$ by $y$ .

$36 y - 36 = 3 \cdot 3 y^{2} + 3 y \cdot - 3 - 3 (3 y) - 3 \cdot - 3$

Multiply $3$ by $3$ .

$36 y - 36 = 9 y^{2} + 3 y \cdot - 3 - 3 (3 y) - 3 \cdot - 3$

Multiply $- 3$ by $3$ .

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

Multiply $3$ by $- 3$ .

$36 y - 36 = 9 y^{2} - 9 y - 9 y - 3 \cdot - 3$

Multiply $- 3$ by $- 3$ .

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

Subtract $9 y$ from $- 9 y$ .

$36 y - 36 = 9 y^{2} - 18 y + 9$

Step 7

Solve for $y$ .

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

$9 y^{2} - 18 y + 9 = 36 y - 36$

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

Subtract $36 y$ from both sides of the equation.

$9 y^{2} - 18 y + 9 - 36 y = - 36$

Subtract $36 y$ from $- 18 y$ .

$9 y^{2} - 54 y + 9 = - 36$

Add $36$ to both sides of the equation.

$9 y^{2} - 54 y + 9 + 36 = 0$

Add $9$ and $36$ .

$9 y^{2} - 54 y + 45 = 0$

Factor the left side of the equation.

Factor $9$ out of $9 y^{2} - 54 y + 45$ .

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

$9 (y^{2}) - 54 y + 45 = 0$

Factor $9$ out of $- 54 y$ .

$9 (y^{2}) + 9 (- 6 y) + 45 = 0$

Factor $9$ out of $45$ .

$9 y^{2} + 9 (- 6 y) + 9 \cdot 5 = 0$

Factor $9$ out of $9 y^{2} + 9 (- 6 y)$ .

$9 (y^{2} - 6 y) + 9 \cdot 5 = 0$

Factor $9$ out of $9 (y^{2} - 6 y) + 9 \cdot 5$ .

$9 (y^{2} - 6 y + 5) = 0$

Factor.

Factor $y^{2} - 6 y + 5$ using the AC method.

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 $5$ and whose sum is $- 6$ .

$- 5, - 1$

Write the factored form using these integers.

$9 ((y - 5) (y - 1)) = 0$

Remove unnecessary parentheses.

$9 (y - 5) (y - 1) = 0$

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

$y - 5 = 0$

$y - 1 = 0$

Set $y - 5$ equal to $0$ and solve for $y$ .

Set $y - 5$ equal to $0$ .

$y - 5 = 0$

Add $5$ to both sides of the equation.

$y = 5$

Set $y - 1$ equal to $0$ and solve for $y$ .

Set $y - 1$ equal to $0$ .

$y - 1 = 0$

Add $1$ to both sides of the equation.

$y = 1$

The final solution is all the values that make $9 (y - 5) (y - 1) = 0$ true.

$y = 5, 1$

Solve for y square root of 4y+5- square root of y-1=3

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