Solve for q q-6 = square root of 27-2q

$q - 6 = \sqrt{27 - 2 q}$

Step 1

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

$\sqrt{27 - 2 q} = q - 6$

Step 2

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

${\sqrt{27 - 2 q}}^{2} = {(q - 6)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{27 - 2 q}$ as ${(27 - 2 q)}^{\frac{1}{2}}$ .

${({(27 - 2 q)}^{\frac{1}{2}})}^{2} = {(q - 6)}^{2}$

Simplify the left side.

Simplify ${({(27 - 2 q)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(27 - 2 q)}^{\frac{1}{2}})}^{2}$ .

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

${(27 - 2 q)}^{\frac{1}{2} \cdot 2} = {(q - 6)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(27 - 2 q)}^{\frac{1}{2} \cdot 2} = {(q - 6)}^{2}$

Rewrite the expression.

${(27 - 2 q)}^{1} = {(q - 6)}^{2}$

Simplify.

$27 - 2 q = {(q - 6)}^{2}$

Simplify the right side.

Simplify ${(q - 6)}^{2}$ .

Rewrite ${(q - 6)}^{2}$ as $(q - 6) (q - 6)$ .

$27 - 2 q = (q - 6) (q - 6)$

Expand $(q - 6) (q - 6)$ using the FOIL Method.

Apply the distributive property.

$27 - 2 q = q (q - 6) - 6 (q - 6)$

Apply the distributive property.

$27 - 2 q = q \cdot q + q \cdot - 6 - 6 (q - 6)$

Apply the distributive property.

$27 - 2 q = q \cdot q + q \cdot - 6 - 6 q - 6 \cdot - 6$

Simplify and combine like terms.

Simplify each term.

Multiply $q$ by $q$ .

$27 - 2 q = q^{2} + q \cdot - 6 - 6 q - 6 \cdot - 6$

Move $- 6$ to the left of $q$ .

$27 - 2 q = q^{2} - 6 \cdot q - 6 q - 6 \cdot - 6$

Multiply $- 6$ by $- 6$ .

$27 - 2 q = q^{2} - 6 q - 6 q + 36$

Subtract $6 q$ from $- 6 q$ .

$27 - 2 q = q^{2} - 12 q + 36$

Step 4

Solve for $q$ .

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

$q^{2} - 12 q + 36 = 27 - 2 q$

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

Add $2 q$ to both sides of the equation.

$q^{2} - 12 q + 36 + 2 q = 27$

Add $- 12 q$ and $2 q$ .

$q^{2} - 10 q + 36 = 27$

Subtract $27$ from both sides of the equation.

$q^{2} - 10 q + 36 - 27 = 0$

Subtract $27$ from $36$ .

$q^{2} - 10 q + 9 = 0$

Factor $q^{2} - 10 q + 9$ 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 $9$ and whose sum is $- 10$ .

$- 9, - 1$

Write the factored form using these integers.

$(q - 9) (q - 1) = 0$

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

$q - 9 = 0$

$q - 1 = 0$

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

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

$q - 9 = 0$

Add $9$ to both sides of the equation.

$q = 9$

Set $q - 1$ equal to $0$ and solve for $q$ .

Set $q - 1$ equal to $0$ .

$q - 1 = 0$

Add $1$ to both sides of the equation.

$q = 1$

The final solution is all the values that make $(q - 9) (q - 1) = 0$ true.

$q = 9, 1$

Step 5

Exclude the solutions that do not make $q - 6 = \sqrt{27 - 2 q}$ true.

$q = 9$

Solve for q q-6 = square root of 27-2q

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