Solve for y square root of y^2-20y+100=y-10

$\sqrt{y^{2} - 20 y + 100} = y - 10$

Step 1

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

${\sqrt{y^{2} - 20 y + 100}}^{2} = {(y - 10)}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y^{2} - 20 y + 100}$ as ${(y^{2} - 20 y + 100)}^{\frac{1}{2}}$ .

${({(y^{2} - 20 y + 100)}^{\frac{1}{2}})}^{2} = {(y - 10)}^{2}$

Simplify the left side.

Simplify ${({(y^{2} - 20 y + 100)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(y^{2} - 20 y + 100)}^{\frac{1}{2}})}^{2}$ .

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

${(y^{2} - 20 y + 100)}^{\frac{1}{2} \cdot 2} = {(y - 10)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(y^{2} - 20 y + 100)}^{\frac{1}{2} \cdot 2} = {(y - 10)}^{2}$

Rewrite the expression.

${(y^{2} - 20 y + 100)}^{1} = {(y - 10)}^{2}$

Simplify.

$y^{2} - 20 y + 100 = {(y - 10)}^{2}$

Simplify the right side.

Simplify ${(y - 10)}^{2}$ .

Rewrite ${(y - 10)}^{2}$ as $(y - 10) (y - 10)$ .

$y^{2} - 20 y + 100 = (y - 10) (y - 10)$

Expand $(y - 10) (y - 10)$ using the FOIL Method.

Apply the distributive property.

$y^{2} - 20 y + 100 = y (y - 10) - 10 (y - 10)$

Apply the distributive property.

$y^{2} - 20 y + 100 = y \cdot y + y \cdot - 10 - 10 (y - 10)$

Apply the distributive property.

$y^{2} - 20 y + 100 = y \cdot y + y \cdot - 10 - 10 y - 10 \cdot - 10$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$y^{2} - 20 y + 100 = y^{2} + y \cdot - 10 - 10 y - 10 \cdot - 10$

Move $- 10$ to the left of $y$ .

$y^{2} - 20 y + 100 = y^{2} - 10 \cdot y - 10 y - 10 \cdot - 10$

Multiply $- 10$ by $- 10$ .

$y^{2} - 20 y + 100 = y^{2} - 10 y - 10 y + 100$

Subtract $10 y$ from $- 10 y$ .

$y^{2} - 20 y + 100 = y^{2} - 20 y + 100$

Step 3

Solve for $y$ .

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

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

$y^{2} - 20 y + 100 - y^{2} = - 20 y + 100$

Add $20 y$ to both sides of the equation.

$y^{2} - 20 y + 100 - y^{2} + 20 y = 100$

Combine the opposite terms in $y^{2} - 20 y + 100 - y^{2} + 20 y$ .

Subtract $y^{2}$ from $y^{2}$ .

$- 20 y + 100 + 0 + 20 y = 100$

Add $- 20 y + 100$ and $0$ .

$- 20 y + 100 + 20 y = 100$

Add $- 20 y$ and $20 y$ .

$0 + 100 = 100$

Add $0$ and $100$ .

$100 = 100$

Since $100 = 100$ , the equation will always be true for any value of $y$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Solve for y square root of y^2-20y+100=y-10

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