Solve by Factoring y+1 = square root of 12y-23

$y + 1 = \sqrt{12 y - 23}$

Step 1

Subtract $\sqrt{12 y - 23}$ from both sides of the equation.

$y + 1 - \sqrt{12 y - 23} = 0$

Step 2

Move all terms not containing $- \sqrt{12 y - 23}$ to the right side of the equation.

Subtract $y$ from both sides of the equation.

$1 - \sqrt{12 y - 23} = - y$

Subtract $1$ from both sides of the equation.

$- \sqrt{12 y - 23} = - y - 1$

Step 3

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

${(- \sqrt{12 y - 23})}^{2} = {(- y - 1)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{12 y - 23}$ as ${(12 y - 23)}^{\frac{1}{2}}$ .

${(- {(12 y - 23)}^{\frac{1}{2}})}^{2} = {(- y - 1)}^{2}$

Simplify the left side.

Simplify ${(- {(12 y - 23)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- {(12 y - 23)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(12 y - 23)}^{\frac{1}{2}})}^{2} = {(- y - 1)}^{2}$

Raise $- 1$ to the power of $2$ .

$1 {({(12 y - 23)}^{\frac{1}{2}})}^{2} = {(- y - 1)}^{2}$

Multiply ${({(12 y - 23)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(12 y - 23)}^{\frac{1}{2}})}^{2} = {(- y - 1)}^{2}$

Multiply the exponents in ${({(12 y - 23)}^{\frac{1}{2}})}^{2}$ .

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

${(12 y - 23)}^{\frac{1}{2} \cdot 2} = {(- y - 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(12 y - 23)}^{\frac{1}{2} \cdot 2} = {(- y - 1)}^{2}$

Rewrite the expression.

${(12 y - 23)}^{1} = {(- y - 1)}^{2}$

Simplify.

$12 y - 23 = {(- y - 1)}^{2}$

Simplify the right side.

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

Rewrite ${(- y - 1)}^{2}$ as $(- y - 1) (- y - 1)$ .

$12 y - 23 = (- y - 1) (- y - 1)$

Expand $(- y - 1) (- y - 1)$ using the FOIL Method.

Apply the distributive property.

$12 y - 23 = - y (- y - 1) - 1 (- y - 1)$

Apply the distributive property.

$12 y - 23 = - y (- y) - y \cdot - 1 - 1 (- y - 1)$

Apply the distributive property.

$12 y - 23 = - y (- y) - y \cdot - 1 - 1 (- y) - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$12 y - 23 = - 1 \cdot - 1 y \cdot y - y \cdot - 1 - 1 (- y) - 1 \cdot - 1$

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

Move $y$ .

$12 y - 23 = - 1 \cdot - 1 (y \cdot y) - y \cdot - 1 - 1 (- y) - 1 \cdot - 1$

Multiply $y$ by $y$ .

$12 y - 23 = - 1 \cdot - 1 y^{2} - y \cdot - 1 - 1 (- y) - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$12 y - 23 = 1 y^{2} - y \cdot - 1 - 1 (- y) - 1 \cdot - 1$

Multiply $y^{2}$ by $1$ .

$12 y - 23 = y^{2} - y \cdot - 1 - 1 (- y) - 1 \cdot - 1$

Multiply $- y \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

$12 y - 23 = y^{2} + 1 y - 1 (- y) - 1 \cdot - 1$

Multiply $y$ by $1$ .

$12 y - 23 = y^{2} + y - 1 (- y) - 1 \cdot - 1$

Multiply $- 1 (- y)$ .

Multiply $- 1$ by $- 1$ .

$12 y - 23 = y^{2} + y + 1 y - 1 \cdot - 1$

Multiply $y$ by $1$ .

$12 y - 23 = y^{2} + y + y - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$12 y - 23 = y^{2} + y + y + 1$

Add $y$ and $y$ .

$12 y - 23 = y^{2} + 2 y + 1$

Step 5

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.

$y^{2} + 2 y + 1 = 12 y - 23$

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

Subtract $12 y$ from both sides of the equation.

$y^{2} + 2 y + 1 - 12 y = - 23$

Subtract $12 y$ from $2 y$ .

$y^{2} - 10 y + 1 = - 23$

Add $23$ to both sides of the equation.

$y^{2} - 10 y + 1 + 23 = 0$

Add $1$ and $23$ .

$y^{2} - 10 y + 24 = 0$

Factor $y^{2} - 10 y + 24$ 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 $24$ and whose sum is $- 10$ .

$- 6, - 4$

Write the factored form using these integers.

$(y - 6) (y - 4) = 0$

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

$y - 6 = 0$

$y - 4 = 0$

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 $y - 4$ 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 $(y - 6) (y - 4) = 0$ true.

$y = 6, 4$

Solve by Factoring y+1 = square root of 12y-23

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