Precalculus Examples

$x - 12 \sqrt{x} + 36 = 0$

Step 1

Move all terms not containing $- 12 \sqrt{x}$ to the right side of the equation.

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Step 1.1

Subtract $x$ from both sides of the equation.

$- 12 \sqrt{x} + 36 = - x$

Step 1.2

Subtract $36$ from both sides of the equation.

$- 12 \sqrt{x} = - x - 36$

Step 2

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

${(- 12 \sqrt{x})}^{2} = {(- x - 36)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

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

${(- 12 x^{\frac{1}{2}})}^{2} = {(- x - 36)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(- 12 x^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $- 12 x^{\frac{1}{2}}$ .

${(- 12)}^{2} {(x^{\frac{1}{2}})}^{2} = {(- x - 36)}^{2}$

Step 3.2.1.2

Raise $- 12$ to the power of $2$ .

$144 {(x^{\frac{1}{2}})}^{2} = {(- x - 36)}^{2}$

Step 3.2.1.3

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

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Step 3.2.1.3.1

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

$144 x^{\frac{1}{2} \cdot 2} = {(- x - 36)}^{2}$

Step 3.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.1.3.2.1

Cancel the common factor.

$144 x^{\frac{1}{2} \cdot 2} = {(- x - 36)}^{2}$

Step 3.2.1.3.2.2

Rewrite the expression.

$144 x^{1} = {(- x - 36)}^{2}$

Step 3.2.1.4

Simplify.

$144 x = {(- x - 36)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(- x - 36)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(- x - 36)}^{2}$ as $(- x - 36) (- x - 36)$ .

$144 x = (- x - 36) (- x - 36)$

Step 3.3.1.2

Expand $(- x - 36) (- x - 36)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$144 x = - x (- x - 36) - 36 (- x - 36)$

Step 3.3.1.2.2

Apply the distributive property.

$144 x = - x (- x) - x \cdot - 36 - 36 (- x - 36)$

Step 3.3.1.2.3

Apply the distributive property.

$144 x = - x (- x) - x \cdot - 36 - 36 (- x) - 36 \cdot - 36$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$144 x = - 1 \cdot - 1 x \cdot x - x \cdot - 36 - 36 (- x) - 36 \cdot - 36$

Step 3.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.1.3.1.2.1

Move $x$ .

$144 x = - 1 \cdot - 1 (x \cdot x) - x \cdot - 36 - 36 (- x) - 36 \cdot - 36$

Step 3.3.1.3.1.2.2

Multiply $x$ by $x$ .

$144 x = - 1 \cdot - 1 x^{2} - x \cdot - 36 - 36 (- x) - 36 \cdot - 36$

Step 3.3.1.3.1.3

Multiply $- 1$ by $- 1$ .

$144 x = 1 x^{2} - x \cdot - 36 - 36 (- x) - 36 \cdot - 36$

Step 3.3.1.3.1.4

Multiply $x^{2}$ by $1$ .

$144 x = x^{2} - x \cdot - 36 - 36 (- x) - 36 \cdot - 36$

Step 3.3.1.3.1.5

Multiply $- 36$ by $- 1$ .

$144 x = x^{2} + 36 x - 36 (- x) - 36 \cdot - 36$

Step 3.3.1.3.1.6

Multiply $- 1$ by $- 36$ .

$144 x = x^{2} + 36 x + 36 x - 36 \cdot - 36$

Step 3.3.1.3.1.7

Multiply $- 36$ by $- 36$ .

$144 x = x^{2} + 36 x + 36 x + 1296$

Step 3.3.1.3.2

Add $36 x$ and $36 x$ .

$144 x = x^{2} + 72 x + 1296$

Step 4

Solve for $x$ .

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Step 4.1

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

$x^{2} + 72 x + 1296 = 144 x$

Step 4.2

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

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Step 4.2.1

Subtract $144 x$ from both sides of the equation.

$x^{2} + 72 x + 1296 - 144 x = 0$

Step 4.2.2

Subtract $144 x$ from $72 x$ .

$x^{2} - 72 x + 1296 = 0$

Step 4.3

Factor using the perfect square rule.

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Step 4.3.1

Rewrite $1296$ as $36^{2}$ .

$x^{2} - 72 x + 36^{2} = 0$

Step 4.3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$72 x = 2 \cdot x \cdot 36$

Step 4.3.3

Rewrite the polynomial.

$x^{2} - 2 \cdot x \cdot 36 + 36^{2} = 0$

Step 4.3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 36$ .

${(x - 36)}^{2} = 0$

Step 4.4

Set the $x - 36$ equal to $0$ .

$x - 36 = 0$

Step 4.5

Add $36$ to both sides of the equation.

$x = 36$