Solve for y square root of 5y+29=6+ square root of y-3

$\sqrt{5 y + 29} = 6 + \sqrt{y - 3}$

Step 1

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

${\sqrt{5 y + 29}}^{2} = {(6 + \sqrt{y - 3})}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 y + 29}$ as ${(5 y + 29)}^{\frac{1}{2}}$ .

${({(5 y + 29)}^{\frac{1}{2}})}^{2} = {(6 + \sqrt{y - 3})}^{2}$

Simplify the left side.

Simplify ${({(5 y + 29)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(5 y + 29)}^{\frac{1}{2}})}^{2}$ .

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

${(5 y + 29)}^{\frac{1}{2} \cdot 2} = {(6 + \sqrt{y - 3})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(5 y + 29)}^{\frac{1}{2} \cdot 2} = {(6 + \sqrt{y - 3})}^{2}$

Rewrite the expression.

${(5 y + 29)}^{1} = {(6 + \sqrt{y - 3})}^{2}$

Simplify.

$5 y + 29 = {(6 + \sqrt{y - 3})}^{2}$

Simplify the right side.

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

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

$5 y + 29 = (6 + \sqrt{y - 3}) (6 + \sqrt{y - 3})$

Expand $(6 + \sqrt{y - 3}) (6 + \sqrt{y - 3})$ using the FOIL Method.

Apply the distributive property.

$5 y + 29 = 6 (6 + \sqrt{y - 3}) + \sqrt{y - 3} (6 + \sqrt{y - 3})$

Apply the distributive property.

$5 y + 29 = 6 \cdot 6 + 6 \sqrt{y - 3} + \sqrt{y - 3} (6 + \sqrt{y - 3})$

Apply the distributive property.

$5 y + 29 = 6 \cdot 6 + 6 \sqrt{y - 3} + \sqrt{y - 3} \cdot 6 + \sqrt{y - 3} \sqrt{y - 3}$

Simplify and combine like terms.

Simplify each term.

Multiply $6$ by $6$ .

$5 y + 29 = 36 + 6 \sqrt{y - 3} + \sqrt{y - 3} \cdot 6 + \sqrt{y - 3} \sqrt{y - 3}$

Move $6$ to the left of $\sqrt{y - 3}$ .

$5 y + 29 = 36 + 6 \sqrt{y - 3} + 6 \cdot \sqrt{y - 3} + \sqrt{y - 3} \sqrt{y - 3}$

Multiply $\sqrt{y - 3} \sqrt{y - 3}$ .

Raise $\sqrt{y - 3}$ to the power of $1$ .

$5 y + 29 = 36 + 6 \sqrt{y - 3} + 6 \sqrt{y - 3} + {\sqrt{y - 3}}^{1} \sqrt{y - 3}$

Raise $\sqrt{y - 3}$ to the power of $1$ .

$5 y + 29 = 36 + 6 \sqrt{y - 3} + 6 \sqrt{y - 3} + {\sqrt{y - 3}}^{1} {\sqrt{y - 3}}^{1}$

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

$5 y + 29 = 36 + 6 \sqrt{y - 3} + 6 \sqrt{y - 3} + {\sqrt{y - 3}}^{1 + 1}$

Add $1$ and $1$ .

$5 y + 29 = 36 + 6 \sqrt{y - 3} + 6 \sqrt{y - 3} + {\sqrt{y - 3}}^{2}$

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y - 3}$ as ${(y - 3)}^{\frac{1}{2}}$ .

$5 y + 29 = 36 + 6 \sqrt{y - 3} + 6 \sqrt{y - 3} + {({(y - 3)}^{\frac{1}{2}})}^{2}$

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

$5 y + 29 = 36 + 6 \sqrt{y - 3} + 6 \sqrt{y - 3} + {(y - 3)}^{\frac{1}{2} \cdot 2}$

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

$5 y + 29 = 36 + 6 \sqrt{y - 3} + 6 \sqrt{y - 3} + {(y - 3)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$5 y + 29 = 36 + 6 \sqrt{y - 3} + 6 \sqrt{y - 3} + {(y - 3)}^{\frac{2}{2}}$

Rewrite the expression.

$5 y + 29 = 36 + 6 \sqrt{y - 3} + 6 \sqrt{y - 3} + {(y - 3)}^{1}$

Simplify.

$5 y + 29 = 36 + 6 \sqrt{y - 3} + 6 \sqrt{y - 3} + y - 3$

Subtract $3$ from $36$ .

$5 y + 29 = 33 + 6 \sqrt{y - 3} + 6 \sqrt{y - 3} + y$

Add $6 \sqrt{y - 3}$ and $6 \sqrt{y - 3}$ .

$5 y + 29 = 33 + 12 \sqrt{y - 3} + y$

Step 3

Solve for $12 \sqrt{y - 3}$ .

Rewrite the equation as $33 + 12 \sqrt{y - 3} + y = 5 y + 29$ .

$33 + 12 \sqrt{y - 3} + y = 5 y + 29$

Move all terms not containing $12 \sqrt{y - 3}$ to the right side of the equation.

Subtract $33$ from both sides of the equation.

$12 \sqrt{y - 3} + y = 5 y + 29 - 33$

Subtract $y$ from both sides of the equation.

$12 \sqrt{y - 3} = 5 y + 29 - 33 - y$

Subtract $y$ from $5 y$ .

$12 \sqrt{y - 3} = 4 y + 29 - 33$

Subtract $33$ from $29$ .

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

Step 4

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

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

Step 5

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y - 3}$ as ${(y - 3)}^{\frac{1}{2}}$ .

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

Simplify the left side.

Simplify ${(12 {(y - 3)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $12 {(y - 3)}^{\frac{1}{2}}$ .

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

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

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

Multiply the exponents in ${({(y - 3)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

Multiply $144$ by $- 3$ .

$144 y - 432 = {(4 y - 4)}^{2}$

Simplify the right side.

Simplify ${(4 y - 4)}^{2}$ .

Rewrite ${(4 y - 4)}^{2}$ as $(4 y - 4) (4 y - 4)$ .

$144 y - 432 = (4 y - 4) (4 y - 4)$

Expand $(4 y - 4) (4 y - 4)$ using the FOIL Method.

Apply the distributive property.

$144 y - 432 = 4 y (4 y - 4) - 4 (4 y - 4)$

Apply the distributive property.

$144 y - 432 = 4 y (4 y) + 4 y \cdot - 4 - 4 (4 y - 4)$

Apply the distributive property.

$144 y - 432 = 4 y (4 y) + 4 y \cdot - 4 - 4 (4 y) - 4 \cdot - 4$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$144 y - 432 = 4 \cdot 4 y \cdot y + 4 y \cdot - 4 - 4 (4 y) - 4 \cdot - 4$

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

Move $y$ .

$144 y - 432 = 4 \cdot 4 (y \cdot y) + 4 y \cdot - 4 - 4 (4 y) - 4 \cdot - 4$

Multiply $y$ by $y$ .

$144 y - 432 = 4 \cdot 4 y^{2} + 4 y \cdot - 4 - 4 (4 y) - 4 \cdot - 4$

Multiply $4$ by $4$ .

$144 y - 432 = 16 y^{2} + 4 y \cdot - 4 - 4 (4 y) - 4 \cdot - 4$

Multiply $- 4$ by $4$ .

$144 y - 432 = 16 y^{2} - 16 y - 4 (4 y) - 4 \cdot - 4$

Multiply $4$ by $- 4$ .

$144 y - 432 = 16 y^{2} - 16 y - 16 y - 4 \cdot - 4$

Multiply $- 4$ by $- 4$ .

$144 y - 432 = 16 y^{2} - 16 y - 16 y + 16$

Subtract $16 y$ from $- 16 y$ .

$144 y - 432 = 16 y^{2} - 32 y + 16$

Step 6

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.

$16 y^{2} - 32 y + 16 = 144 y - 432$

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

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

$16 y^{2} - 32 y + 16 - 144 y = - 432$

Subtract $144 y$ from $- 32 y$ .

$16 y^{2} - 176 y + 16 = - 432$

Add $432$ to both sides of the equation.

$16 y^{2} - 176 y + 16 + 432 = 0$

Add $16$ and $432$ .

$16 y^{2} - 176 y + 448 = 0$

Factor the left side of the equation.

Factor $16$ out of $16 y^{2} - 176 y + 448$ .

Factor $16$ out of $16 y^{2}$ .

$16 (y^{2}) - 176 y + 448 = 0$

Factor $16$ out of $- 176 y$ .

$16 (y^{2}) + 16 (- 11 y) + 448 = 0$

Factor $16$ out of $448$ .

$16 y^{2} + 16 (- 11 y) + 16 \cdot 28 = 0$

Factor $16$ out of $16 y^{2} + 16 (- 11 y)$ .

$16 (y^{2} - 11 y) + 16 \cdot 28 = 0$

Factor $16$ out of $16 (y^{2} - 11 y) + 16 \cdot 28$ .

$16 (y^{2} - 11 y + 28) = 0$

Factor.

Factor $y^{2} - 11 y + 28$ 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 $28$ and whose sum is $- 11$ .

$- 7, - 4$

Write the factored form using these integers.

$16 ((y - 7) (y - 4)) = 0$

Remove unnecessary parentheses.

$16 (y - 7) (y - 4) = 0$

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

$y - 7 = 0$

$y - 4 = 0$

Set $y - 7$ equal to $0$ and solve for $y$ .

Set $y - 7$ equal to $0$ .

$y - 7 = 0$

Add $7$ to both sides of the equation.

$y = 7$

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

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

$y - 4 = 0$

Add $4$ to both sides of the equation.

$y = 4$

The final solution is all the values that make $16 (y - 7) (y - 4) = 0$ true.

$y = 7, 4$

Solve for y square root of 5y+29=6+ square root of y-3

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