Solve by Factoring square root of 144x=x+20

$\sqrt{144 x} = x + 20$

Step 1

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

Subtract $x$ from both sides of the equation.

$\sqrt{144 x} - x = 20$

Subtract $20$ from both sides of the equation.

$\sqrt{144 x} - x - 20 = 0$

Step 2

Simplify each term.

Rewrite $144$ as $12^{2}$ .

$\sqrt{12^{2} x} - x - 20 = 0$

Pull terms out from under the radical.

$12 \sqrt{x} - x - 20 = 0$

Step 3

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

$12 x^{\frac{1}{2}} - x - 20 = 0$

Step 4

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

$12 x^{\frac{1}{2}} - {(x^{\frac{1}{2}})}^{2} - 20 = 0$

Step 5

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

$12 u - u^{2} - 20 = 0$

Step 6

Factor by grouping.

Reorder terms.

$- u^{2} + 12 u - 20 = 0$

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 $a \cdot c = - 1 \cdot - 20 = 20$ and whose sum is $b = 12$ .

Factor $12$ out of $12 u$ .

$- u^{2} + 12 (u) - 20 = 0$

Rewrite $12$ as $2$ plus $10$

$- u^{2} + (2 + 10) u - 20 = 0$

Apply the distributive property.

$- u^{2} + 2 u + 10 u - 20 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(- u^{2} + 2 u) + 10 u - 20 = 0$

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

$u (- u + 2) - 10 (- u + 2) = 0$

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

$(- u + 2) (u - 10) = 0$

Step 7

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

$(- x^{\frac{1}{2}} + 2) (x^{\frac{1}{2}} - 10) = 0$

Step 8

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

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

$x^{\frac{1}{2}} - 10 = 0$

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

Apply the product rule to $- x^{\frac{1}{2}}$ .

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

Raise $- 1$ to the power of $2$ .

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

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

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

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 $x^{\frac{1}{2}} - 10$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{2}} - 10$ equal to $0$ .

$x^{\frac{1}{2}} - 10 = 0$

Solve $x^{\frac{1}{2}} - 10 = 0$ for $x$ .

Add $10$ to both sides of the equation.

$x^{\frac{1}{2}} = 10$

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{2} \cdot 2} = 10^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 10^{2}$

Rewrite the expression.

$x^{1} = 10^{2}$

Simplify.

$x = 10^{2}$

Simplify the right side.

Raise $10$ to the power of $2$ .

$x = 100$

Step 11

The final solution is all the values that make $(- x^{\frac{1}{2}} + 2) (x^{\frac{1}{2}} - 10) = 0$ true.

$x = 4, 100$

Solve by Factoring square root of 144x=x+20

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