Solve for y square root of 4y+1- square root of y-2=3

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

Step 1

Add $\sqrt{y - 2}$ to both sides of the equation.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

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

Move $3$ to the left of $\sqrt{y - 2}$ .

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

Multiply $\sqrt{y - 2} \sqrt{y - 2}$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Subtract $2$ from $9$ .

$4 y + 1 = 7 + 3 \sqrt{y - 2} + 3 \sqrt{y - 2} + y$

Add $3 \sqrt{y - 2}$ and $3 \sqrt{y - 2}$ .

$4 y + 1 = 7 + 6 \sqrt{y - 2} + y$

Step 4

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

Rewrite the equation as $7 + 6 \sqrt{y - 2} + y = 4 y + 1$ .

$7 + 6 \sqrt{y - 2} + y = 4 y + 1$

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

Subtract $7$ from both sides of the equation.

$6 \sqrt{y - 2} + y = 4 y + 1 - 7$

Subtract $y$ from both sides of the equation.

$6 \sqrt{y - 2} = 4 y + 1 - 7 - y$

Subtract $y$ from $4 y$ .

$6 \sqrt{y - 2} = 3 y + 1 - 7$

Subtract $7$ from $1$ .

$6 \sqrt{y - 2} = 3 y - 6$

Step 5

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

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

Step 6

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

$36 {({(y - 2)}^{\frac{1}{2}})}^{2} = {(3 y - 6)}^{2}$

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

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

$36 {(y - 2)}^{\frac{1}{2} \cdot 2} = {(3 y - 6)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$36 {(y - 2)}^{\frac{1}{2} \cdot 2} = {(3 y - 6)}^{2}$

Rewrite the expression.

$36 {(y - 2)}^{1} = {(3 y - 6)}^{2}$

Simplify.

$36 (y - 2) = {(3 y - 6)}^{2}$

Apply the distributive property.

$36 y + 36 \cdot - 2 = {(3 y - 6)}^{2}$

Multiply $36$ by $- 2$ .

$36 y - 72 = {(3 y - 6)}^{2}$

Simplify the right side.

Simplify ${(3 y - 6)}^{2}$ .

Rewrite ${(3 y - 6)}^{2}$ as $(3 y - 6) (3 y - 6)$ .

$36 y - 72 = (3 y - 6) (3 y - 6)$

Expand $(3 y - 6) (3 y - 6)$ using the FOIL Method.

Apply the distributive property.

$36 y - 72 = 3 y (3 y - 6) - 6 (3 y - 6)$

Apply the distributive property.

$36 y - 72 = 3 y (3 y) + 3 y \cdot - 6 - 6 (3 y - 6)$

Apply the distributive property.

$36 y - 72 = 3 y (3 y) + 3 y \cdot - 6 - 6 (3 y) - 6 \cdot - 6$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$36 y - 72 = 3 \cdot 3 y \cdot y + 3 y \cdot - 6 - 6 (3 y) - 6 \cdot - 6$

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

Move $y$ .

$36 y - 72 = 3 \cdot 3 (y \cdot y) + 3 y \cdot - 6 - 6 (3 y) - 6 \cdot - 6$

Multiply $y$ by $y$ .

$36 y - 72 = 3 \cdot 3 y^{2} + 3 y \cdot - 6 - 6 (3 y) - 6 \cdot - 6$

Multiply $3$ by $3$ .

$36 y - 72 = 9 y^{2} + 3 y \cdot - 6 - 6 (3 y) - 6 \cdot - 6$

Multiply $- 6$ by $3$ .

$36 y - 72 = 9 y^{2} - 18 y - 6 (3 y) - 6 \cdot - 6$

Multiply $3$ by $- 6$ .

$36 y - 72 = 9 y^{2} - 18 y - 18 y - 6 \cdot - 6$

Multiply $- 6$ by $- 6$ .

$36 y - 72 = 9 y^{2} - 18 y - 18 y + 36$

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

$36 y - 72 = 9 y^{2} - 36 y + 36$

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.

$9 y^{2} - 36 y + 36 = 36 y - 72$

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

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

$9 y^{2} - 36 y + 36 - 36 y = - 72$

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

$9 y^{2} - 72 y + 36 = - 72$

Add $72$ to both sides of the equation.

$9 y^{2} - 72 y + 36 + 72 = 0$

Add $36$ and $72$ .

$9 y^{2} - 72 y + 108 = 0$

Factor the left side of the equation.

Factor $9$ out of $9 y^{2} - 72 y + 108$ .

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

$9 (y^{2}) - 72 y + 108 = 0$

Factor $9$ out of $- 72 y$ .

$9 (y^{2}) + 9 (- 8 y) + 108 = 0$

Factor $9$ out of $108$ .

$9 y^{2} + 9 (- 8 y) + 9 \cdot 12 = 0$

Factor $9$ out of $9 y^{2} + 9 (- 8 y)$ .

$9 (y^{2} - 8 y) + 9 \cdot 12 = 0$

Factor $9$ out of $9 (y^{2} - 8 y) + 9 \cdot 12$ .

$9 (y^{2} - 8 y + 12) = 0$

Factor.

Factor $y^{2} - 8 y + 12$ 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 $12$ and whose sum is $- 8$ .

$- 6, - 2$

Write the factored form using these integers.

$9 ((y - 6) (y - 2)) = 0$

Remove unnecessary parentheses.

$9 (y - 6) (y - 2) = 0$

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

$y - 6 = 0$

$y - 2 = 0$

Set $y - 6$ equal to $0$ and solve for $y$ .

Set $y - 6$ equal to $0$ .

$y - 6 = 0$

Add $6$ to both sides of the equation.

$y = 6$

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

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

$y - 2 = 0$

Add $2$ to both sides of the equation.

$y = 2$

The final solution is all the values that make $9 (y - 6) (y - 2) = 0$ true.

$y = 6, 2$

Solve for y square root of 4y+1- square root of y-2=3

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