Solve Using the Quadratic Formula x-12 square root of x+27=0

$x - 12 \sqrt{x} + 27 = 0$

Step 1

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

Subtract $x$ from both sides of the equation.

$- 12 \sqrt{x} + 27 = - x$

Subtract $27$ from both sides of the equation.

$- 12 \sqrt{x} = - x - 27$

Step 2

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

${(- 12 \sqrt{x})}^{2} = {(- x - 27)}^{2}$

Step 3

Simplify each side of the equation.

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

${(- 12 x^{\frac{1}{2}})}^{2} = {(- x - 27)}^{2}$

Simplify the left side.

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

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

${(- 12)}^{2} {(x^{\frac{1}{2}})}^{2} = {(- x - 27)}^{2}$

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

$144 {(x^{\frac{1}{2}})}^{2} = {(- x - 27)}^{2}$

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

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

$144 x^{\frac{1}{2} \cdot 2} = {(- x - 27)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$144 x^{\frac{1}{2} \cdot 2} = {(- x - 27)}^{2}$

Rewrite the expression.

$144 x^{1} = {(- x - 27)}^{2}$

Simplify.

$144 x = {(- x - 27)}^{2}$

Simplify the right side.

Simplify ${(- x - 27)}^{2}$ .

Rewrite ${(- x - 27)}^{2}$ as $(- x - 27) (- x - 27)$ .

$144 x = (- x - 27) (- x - 27)$

Expand $(- x - 27) (- x - 27)$ using the FOIL Method.

Apply the distributive property.

$144 x = - x (- x - 27) - 27 (- x - 27)$

Apply the distributive property.

$144 x = - x (- x) - x \cdot - 27 - 27 (- x - 27)$

Apply the distributive property.

$144 x = - x (- x) - x \cdot - 27 - 27 (- x) - 27 \cdot - 27$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$144 x = - 1 \cdot - 1 x \cdot x - x \cdot - 27 - 27 (- x) - 27 \cdot - 27$

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

Move $x$ .

$144 x = - 1 \cdot - 1 (x \cdot x) - x \cdot - 27 - 27 (- x) - 27 \cdot - 27$

Multiply $x$ by $x$ .

$144 x = - 1 \cdot - 1 x^{2} - x \cdot - 27 - 27 (- x) - 27 \cdot - 27$

Multiply $- 1$ by $- 1$ .

$144 x = 1 x^{2} - x \cdot - 27 - 27 (- x) - 27 \cdot - 27$

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

$144 x = x^{2} - x \cdot - 27 - 27 (- x) - 27 \cdot - 27$

Multiply $- 27$ by $- 1$ .

$144 x = x^{2} + 27 x - 27 (- x) - 27 \cdot - 27$

Multiply $- 1$ by $- 27$ .

$144 x = x^{2} + 27 x + 27 x - 27 \cdot - 27$

Multiply $- 27$ by $- 27$ .

$144 x = x^{2} + 27 x + 27 x + 729$

Add $27 x$ and $27 x$ .

$144 x = x^{2} + 54 x + 729$

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.

$x^{2} + 54 x + 729 = 144 x$

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

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

$x^{2} + 54 x + 729 - 144 x = 0$

Subtract $144 x$ from $54 x$ .

$x^{2} - 90 x + 729 = 0$

Factor $x^{2} - 90 x + 729$ 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 $729$ and whose sum is $- 90$ .

$- 81, - 9$

Write the factored form using these integers.

$(x - 81) (x - 9) = 0$

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

$x - 81 = 0$

$x - 9 = 0$

Set $x - 81$ equal to $0$ and solve for $x$ .

Set $x - 81$ equal to $0$ .

$x - 81 = 0$

Add $81$ to both sides of the equation.

$x = 81$

Set $x - 9$ equal to $0$ and solve for $x$ .

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

The final solution is all the values that make $(x - 81) (x - 9) = 0$ true.

$x = 81, 9$

Solve Using the Quadratic Formula x-12 square root of x+27=0

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