Solve for x square root of 25-10x+x^2=x-5

$\sqrt{25 - 10 x + x^{2}} = x - 5$

Step 1

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

${\sqrt{25 - 10 x + x^{2}}}^{2} = {(x - 5)}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{25 - 10 x + x^{2}}$ as ${(25 - 10 x + x^{2})}^{\frac{1}{2}}$ .

${({(25 - 10 x + x^{2})}^{\frac{1}{2}})}^{2} = {(x - 5)}^{2}$

Simplify the left side.

Simplify ${({(25 - 10 x + x^{2})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(25 - 10 x + x^{2})}^{\frac{1}{2}})}^{2}$ .

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

${(25 - 10 x + x^{2})}^{\frac{1}{2} \cdot 2} = {(x - 5)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(25 - 10 x + x^{2})}^{\frac{1}{2} \cdot 2} = {(x - 5)}^{2}$

Rewrite the expression.

${(25 - 10 x + x^{2})}^{1} = {(x - 5)}^{2}$

Simplify.

$25 - 10 x + x^{2} = {(x - 5)}^{2}$

Simplify the right side.

Simplify ${(x - 5)}^{2}$ .

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$25 - 10 x + x^{2} = (x - 5) (x - 5)$

Expand $(x - 5) (x - 5)$ using the FOIL Method.

Apply the distributive property.

$25 - 10 x + x^{2} = x (x - 5) - 5 (x - 5)$

Apply the distributive property.

$25 - 10 x + x^{2} = x \cdot x + x \cdot - 5 - 5 (x - 5)$

Apply the distributive property.

$25 - 10 x + x^{2} = x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$25 - 10 x + x^{2} = x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5$

Move $- 5$ to the left of $x$ .

$25 - 10 x + x^{2} = x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5$

Multiply $- 5$ by $- 5$ .

$25 - 10 x + x^{2} = x^{2} - 5 x - 5 x + 25$

Subtract $5 x$ from $- 5 x$ .

$25 - 10 x + x^{2} = x^{2} - 10 x + 25$

Step 3

Solve for $x$ .

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

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

$25 - 10 x + x^{2} - x^{2} = - 10 x + 25$

Add $10 x$ to both sides of the equation.

$25 - 10 x + x^{2} - x^{2} + 10 x = 25$

Combine the opposite terms in $25 - 10 x + x^{2} - x^{2} + 10 x$ .

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

$25 - 10 x + 0 + 10 x = 25$

Add $25 - 10 x$ and $0$ .

$25 - 10 x + 10 x = 25$

Add $- 10 x$ and $10 x$ .

$25 + 0 = 25$

Add $25$ and $0$ .

$25 = 25$

Since $25 = 25$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Solve for x square root of 25-10x+x^2=x-5

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