Solve for y y^2- square root of y^2-|y-2|-11=0

$y^{2} - \sqrt{y^{2}} - | y - 2 | - 11 = 0$

Step 1

Pull terms out from under the radical, assuming positive real numbers.

$y^{2} - y - | y - 2 | - 11 = 0$

Step 2

Move all terms not containing $| y - 2 |$ to the right side of the equation.

Subtract $y^{2}$ from both sides of the equation.

$- y - | y - 2 | - 11 = - y^{2}$

Add $y$ to both sides of the equation.

$- | y - 2 | - 11 = - y^{2} + y$

Add $11$ to both sides of the equation.

$- | y - 2 | = - y^{2} + y + 11$

Step 3

Divide each term in $- | y - 2 | = - y^{2} + y + 11$ by $- 1$ and simplify.

Divide each term in $- | y - 2 | = - y^{2} + y + 11$ by $- 1$ .

$\frac{- | y - 2 |}{- 1} = \frac{- y^{2}}{- 1} + \frac{y}{- 1} + \frac{11}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{| y - 2 |}{1} = \frac{- y^{2}}{- 1} + \frac{y}{- 1} + \frac{11}{- 1}$

Divide $| y - 2 |$ by $1$ .

$| y - 2 | = \frac{- y^{2}}{- 1} + \frac{y}{- 1} + \frac{11}{- 1}$

Simplify the right side.

Simplify each term.

Dividing two negative values results in a positive value.

$| y - 2 | = \frac{y^{2}}{1} + \frac{y}{- 1} + \frac{11}{- 1}$

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

$| y - 2 | = y^{2} + \frac{y}{- 1} + \frac{11}{- 1}$

Move the negative one from the denominator of $\frac{y}{- 1}$ .

$| y - 2 | = y^{2} - 1 \cdot y + \frac{11}{- 1}$

Rewrite $- 1 \cdot y$ as $- y$ .

$| y - 2 | = y^{2} - y + \frac{11}{- 1}$

Divide $11$ by $- 1$ .

$| y - 2 | = y^{2} - y - 11$

Step 4

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$y - 2 = \pm (y^{2} - y - 11)$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$y - 2 = y^{2} - y - 11$

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

$y^{2} - y - 11 = y - 2$

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

Subtract $y$ from both sides of the equation.

$y^{2} - y - 11 - y = - 2$

Subtract $y$ from $- y$ .

$y^{2} - 2 y - 11 = - 2$

Move all terms to the left side of the equation and simplify.

Add $2$ to both sides of the equation.

$y^{2} - 2 y - 11 + 2 = 0$

Add $- 11$ and $2$ .

$y^{2} - 2 y - 9 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = - 2$ , and $c = - 9$ into the quadratic formula and solve for $y$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot - 9)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

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

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 9$ .

Multiply $- 4$ by $1$ .

$y = \frac{2 \pm \sqrt{4 - 4 \cdot - 9}}{2 \cdot 1}$

Multiply $- 4$ by $- 9$ .

$y = \frac{2 \pm \sqrt{4 + 36}}{2 \cdot 1}$

Add $4$ and $36$ .

$y = \frac{2 \pm \sqrt{40}}{2 \cdot 1}$

Rewrite $40$ as $2^{2} \cdot 10$ .

Factor $4$ out of $40$ .

$y = \frac{2 \pm \sqrt{4 (10)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$y = \frac{2 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{2 \pm 2 \sqrt{10}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{2 \pm 2 \sqrt{10}}{2}$

Simplify $\frac{2 \pm 2 \sqrt{10}}{2}$ .

$y = 1 \pm \sqrt{10}$

The final answer is the combination of both solutions.

$y = 1 + \sqrt{10}, 1 - \sqrt{10}$

Next, use the negative value of the $\pm$ to find the second solution.

$y - 2 = - (y^{2} - y - 11)$

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

$- (y^{2} - y - 11) = y - 2$

Simplify $- (y^{2} - y - 11)$ .

Rewrite.

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

Simplify by adding zeros.

$- (y^{2} - y - 11) = y - 2$

Apply the distributive property.

$- y^{2} - - y - - 11 = y - 2$

Simplify.

Multiply $- - y$ .

Multiply $- 1$ by $- 1$ .

$- y^{2} + 1 y - - 11 = y - 2$

Multiply $y$ by $1$ .

$- y^{2} + y - - 11 = y - 2$

Multiply $- 1$ by $- 11$ .

$- y^{2} + y + 11 = y - 2$

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

Subtract $y$ from both sides of the equation.

$- y^{2} + y + 11 - y = - 2$

Combine the opposite terms in $- y^{2} + y + 11 - y$ .

Subtract $y$ from $y$ .

$- y^{2} + 0 + 11 = - 2$

Add $- y^{2}$ and $0$ .

$- y^{2} + 11 = - 2$

Move all terms not containing $y$ to the right side of the equation.

Subtract $11$ from both sides of the equation.

$- y^{2} = - 2 - 11$

Subtract $11$ from $- 2$ .

$- y^{2} = - 13$

Divide each term in $- y^{2} = - 13$ by $- 1$ and simplify.

Divide each term in $- y^{2} = - 13$ by $- 1$ .

$\frac{- y^{2}}{- 1} = \frac{- 13}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y^{2}}{1} = \frac{- 13}{- 1}$

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

$y^{2} = \frac{- 13}{- 1}$

Simplify the right side.

Divide $- 13$ by $- 1$ .

$y^{2} = 13$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{13}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{13}$

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{13}$

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{13}, - \sqrt{13}$

The complete solution is the result of both the positive and negative portions of the solution.

$y = 1 + \sqrt{10}, 1 - \sqrt{10}, \sqrt{13}, - \sqrt{13}$

Step 6

Exclude the solutions that do not make $y^{2} - \sqrt{y^{2}} - | y - 2 | - 11 = 0$ true.

$y = 1 + \sqrt{10}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = 1 + \sqrt{10}$

Decimal Form:

$y = 4.16227766 \dots$

Solve for y y^2- square root of y^2-|y-2|-11=0

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