Solve by Factoring square root of x=x-6

$\sqrt{x} = x - 6$

Step 1

Move all the expressions to the left side of the equation.

Subtract $x$ from both sides of the equation.

$\sqrt{x} - x = - 6$

Add $6$ to both sides of the equation.

$\sqrt{x} - x + 6 = 0$

Step 2

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

$x^{\frac{1}{2}} - x + 6 = 0$

Step 3

Rewrite $x$ as ${(x^{\frac{1}{2}})}^{2}$ .

$x^{\frac{1}{2}} - {(x^{\frac{1}{2}})}^{2} + 6 = 0$

Step 4

Let $u = x^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of $x^{\frac{1}{2}}$ .

$u - u^{2} + 6 = 0$

Step 5

Factor by grouping.

Reorder terms.

$- u^{2} + u + 6 = 0$

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 6 = - 6$ and whose sum is $b = 1$ .

Multiply by $1$ .

$- u^{2} + 1 u + 6 = 0$

Rewrite $1$ as $- 2$ plus $3$

$- u^{2} + (- 2 + 3) u + 6 = 0$

Apply the distributive property.

$- u^{2} - 2 u + 3 u + 6 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(- u^{2} - 2 u) + 3 u + 6 = 0$

Factor out the greatest common factor (GCF) from each group.

$u (- u - 2) - 3 (- u - 2) = 0$

Factor the polynomial by factoring out the greatest common factor, $- u - 2$ .

$(- u - 2) (u - 3) = 0$

Step 6

Replace all occurrences of $u$ with $x^{\frac{1}{2}}$ .

$(- x^{\frac{1}{2}} - 2) (x^{\frac{1}{2}} - 3) = 0$

Step 7

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

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

$(- (x^{\frac{1}{2}}) - 2) (x^{\frac{1}{2}} - 3) = 0$

Rewrite $- 2$ as $- 1 (2)$ .

$(- (x^{\frac{1}{2}}) - 1 \cdot 2) (x^{\frac{1}{2}} - 3) = 0$

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

$- (x^{\frac{1}{2}} + 2) (x^{\frac{1}{2}} - 3) = 0$

Step 8

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

$x^{\frac{1}{2}} + 2 = 0$

$x^{\frac{1}{2}} - 3 = 0$

Step 9

Set $x^{\frac{1}{2}} + 2$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{2}} + 2$ equal to $0$ .

$x^{\frac{1}{2}} + 2 = 0$

Solve $x^{\frac{1}{2}} + 2 = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

$x^{\frac{1}{2}} = - 2$

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Simplify the exponent.

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Raise $- 2$ to the power of $2$ .

$x = 4$

Step 10

Set $x^{\frac{1}{2}} - 3$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{2}} - 3$ equal to $0$ .

$x^{\frac{1}{2}} - 3 = 0$

Solve $x^{\frac{1}{2}} - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$x^{\frac{1}{2}} = 3$

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{2}})}^{2} = 3^{2}$

Simplify the exponent.

Simplify the left side.

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

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

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

$x^{\frac{1}{2} \cdot 2} = 3^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 3^{2}$

Rewrite the expression.

$x^{1} = 3^{2}$

Simplify.

$x = 3^{2}$

Simplify the right side.

Raise $3$ to the power of $2$ .

$x = 9$

Step 11

The final solution is all the values that make $- (x^{\frac{1}{2}} + 2) (x^{\frac{1}{2}} - 3) = 0$ true.

$x = 4, 9$

Step 12

Exclude the solutions that do not make $- (x^{\frac{1}{2}} + 2) (x^{\frac{1}{2}} - 3) = 0$ true.

$x = 9$

Solve by Factoring square root of x=x-6

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