Solve by Factoring square root of 5x-x+10=0

$\sqrt{5 x} - x + 10 = 0$

Step 1

Move all terms not containing $\sqrt{5 x}$ to the right side of the equation.

Add $x$ to both sides of the equation.

$\sqrt{5 x} + 10 = x$

Subtract $10$ from both sides of the equation.

$\sqrt{5 x} = x - 10$

Step 2

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

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

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x}$ as ${(5 x)}^{\frac{1}{2}}$ .

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

Simplify the left side.

Simplify ${({(5 x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(5 x)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

$5 x = (x - 10) (x - 10)$

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

Apply the distributive property.

$5 x = x (x - 10) - 10 (x - 10)$

Apply the distributive property.

$5 x = x \cdot x + x \cdot - 10 - 10 (x - 10)$

Apply the distributive property.

$5 x = x \cdot x + x \cdot - 10 - 10 x - 10 \cdot - 10$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $- 10$ by $- 10$ .

$5 x = x^{2} - 10 x - 10 x + 100$

Subtract $10 x$ from $- 10 x$ .

$5 x = x^{2} - 20 x + 100$

Step 4

Solve for $x$ .

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

$x^{2} - 20 x + 100 = 5 x$

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

Subtract $5 x$ from both sides of the equation.

$x^{2} - 20 x + 100 - 5 x = 0$

Subtract $5 x$ from $- 20 x$ .

$x^{2} - 25 x + 100 = 0$

Factor $x^{2} - 25 x + 100$ 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 $100$ and whose sum is $- 25$ .

$- 20, - 5$

Write the factored form using these integers.

$(x - 20) (x - 5) = 0$

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

$x - 20 = 0$

$x - 5 = 0$

Set $x - 20$ equal to $0$ and solve for $x$ .

Set $x - 20$ equal to $0$ .

$x - 20 = 0$

Add $20$ to both sides of the equation.

$x = 20$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

The final solution is all the values that make $(x - 20) (x - 5) = 0$ true.

$x = 20, 5$

Step 5

Exclude the solutions that do not make $\sqrt{5 x} - x + 10 = 0$ true.

$x = 20$

Solve by Factoring square root of 5x-x+10=0

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