Evaluate y = square root of 2-y

$y = \sqrt{2 - y}$

Step 1

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

$\sqrt{2 - y} = y$

Step 2

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

${\sqrt{2 - y}}^{2} = y^{2}$

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$2 - y = y^{2}$

Step 4

Solve for $y$ .

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

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

Factor the left side of the equation.

Factor $- 1$ out of $2 - y - y^{2}$ .

Reorder the expression.

Move $2$ .

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

Reorder $- y$ and $- y^{2}$ .

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

Factor $- 1$ out of $- y^{2}$ .

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

Factor $- 1$ out of $- y$ .

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

Rewrite $2$ as $- 1 (- 2)$ .

$- (y^{2}) - (y) - 1 \cdot - 2 = 0$

Factor $- 1$ out of $- (y^{2}) - (y)$ .

$- (y^{2} + y) - 1 \cdot - 2 = 0$

Factor $- 1$ out of $- (y^{2} + y) - 1 (- 2)$ .

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

Factor.

Factor $y^{2} + y - 2$ 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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Write the factored form using these integers.

$- ((y - 1) (y + 2)) = 0$

Remove unnecessary parentheses.

$- (y - 1) (y + 2) = 0$

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

$y - 1 = 0$

$y + 2 = 0$

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.

$y = 1$

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

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

$y + 2 = 0$

Subtract $2$ from both sides of the equation.

$y = - 2$

The final solution is all the values that make $- (y - 1) (y + 2) = 0$ true.

$y = 1, - 2$

Step 5

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

$y = 1$

Evaluate y = square root of 2-y

Related Questions

Support LCMGCF.com by disabling your adblocker.