Solve by Factoring square root of 30-x=x

$\sqrt{30 - x} = x$

Step 1

Subtract $x$ from both sides of the equation.

$\sqrt{30 - x} - x = 0$

Step 2

Add $x$ to both sides of the equation.

$\sqrt{30 - x} = x$

Step 3

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

${\sqrt{30 - x}}^{2} = x^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{30 - x}$ as ${(30 - x)}^{\frac{1}{2}}$ .

${({(30 - x)}^{\frac{1}{2}})}^{2} = x^{2}$

Simplify the left side.

Simplify ${({(30 - x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(30 - x)}^{\frac{1}{2}})}^{2}$ .

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

${(30 - x)}^{\frac{1}{2} \cdot 2} = x^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(30 - x)}^{\frac{1}{2} \cdot 2} = x^{2}$

Rewrite the expression.

${(30 - x)}^{1} = x^{2}$

Simplify.

$30 - x = x^{2}$

Step 5

Solve for $x$ .

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

$30 - x - x^{2} = 0$

Factor the left side of the equation.

Factor $- 1$ out of $30 - x - x^{2}$ .

Reorder the expression.

Move $30$ .

$- x - x^{2} + 30 = 0$

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

$- x^{2} - x + 30 = 0$

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

$- (x^{2}) - x + 30 = 0$

Factor $- 1$ out of $- x$ .

$- (x^{2}) - (x) + 30 = 0$

Rewrite $30$ as $- 1 (- 30)$ .

$- (x^{2}) - (x) - 1 \cdot - 30 = 0$

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

$- (x^{2} + x) - 1 \cdot - 30 = 0$

Factor $- 1$ out of $- (x^{2} + x) - 1 (- 30)$ .

$- (x^{2} + x - 30) = 0$

Factor.

Factor $x^{2} + x - 30$ 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 $- 30$ and whose sum is $1$ .

$- 5, 6$

Write the factored form using these integers.

$- ((x - 5) (x + 6)) = 0$

Remove unnecessary parentheses.

$- (x - 5) (x + 6) = 0$

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

$x - 5 = 0$

$x + 6 = 0$

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.

$x = 5$

Set $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

The final solution is all the values that make $- (x - 5) (x + 6) = 0$ true.

$x = 5, - 6$

Step 6

Exclude the solutions that do not make $\sqrt{30 - x} - x = 0$ true.

$x = 5$

Solve by Factoring square root of 30-x=x

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