Solve for y square root of y^2-7y- square root of y+9=0

$\sqrt{y^{2} - 7 y} - \sqrt{y + 9} = 0$

Step 1

Add $\sqrt{y + 9}$ to both sides of the equation.

$\sqrt{y^{2} - 7 y} = \sqrt{y + 9}$

Step 2

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

${\sqrt{y^{2} - 7 y}}^{2} = {\sqrt{y + 9}}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y^{2} - 7 y}$ as ${(y^{2} - 7 y)}^{\frac{1}{2}}$ .

${({(y^{2} - 7 y)}^{\frac{1}{2}})}^{2} = {\sqrt{y + 9}}^{2}$

Simplify the left side.

Simplify ${({(y^{2} - 7 y)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(y^{2} - 7 y)}^{\frac{1}{2}})}^{2}$ .

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

${(y^{2} - 7 y)}^{\frac{1}{2} \cdot 2} = {\sqrt{y + 9}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(y^{2} - 7 y)}^{\frac{1}{2} \cdot 2} = {\sqrt{y + 9}}^{2}$

Rewrite the expression.

${(y^{2} - 7 y)}^{1} = {\sqrt{y + 9}}^{2}$

Simplify.

$y^{2} - 7 y = {\sqrt{y + 9}}^{2}$

Simplify the right side.

Rewrite ${\sqrt{y + 9}}^{2}$ as $y + 9$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y + 9}$ as ${(y + 9)}^{\frac{1}{2}}$ .

$y^{2} - 7 y = {({(y + 9)}^{\frac{1}{2}})}^{2}$

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

$y^{2} - 7 y = {(y + 9)}^{\frac{1}{2} \cdot 2}$

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

$y^{2} - 7 y = {(y + 9)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y^{2} - 7 y = {(y + 9)}^{\frac{2}{2}}$

Rewrite the expression.

$y^{2} - 7 y = {(y + 9)}^{1}$

Simplify.

$y^{2} - 7 y = y + 9$

Step 4

Solve for $y$ .

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

Subtract $y$ from both sides of the equation.

$y^{2} - 7 y - y = 9$

Subtract $y$ from $- 7 y$ .

$y^{2} - 8 y = 9$

Subtract $9$ from both sides of the equation.

$y^{2} - 8 y - 9 = 0$

Factor $y^{2} - 8 y - 9$ 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 $- 9$ and whose sum is $- 8$ .

$- 9, 1$

Write the factored form using these integers.

$(y - 9) (y + 1) = 0$

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

$y - 9 = 0$

$y + 1 = 0$

Set $y - 9$ equal to $0$ and solve for $y$ .

Set $y - 9$ equal to $0$ .

$y - 9 = 0$

Add $9$ to both sides of the equation.

$y = 9$

Set $y + 1$ equal to $0$ and solve for $y$ .

Set $y + 1$ equal to $0$ .

$y + 1 = 0$

Subtract $1$ from both sides of the equation.

$y = - 1$

The final solution is all the values that make $(y - 9) (y + 1) = 0$ true.

$y = 9, - 1$

Solve for y square root of y^2-7y- square root of y+9=0

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