Solve for q square root of 24+8q=q+3

$\sqrt{24 + 8 q} = q + 3$

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

${\sqrt{24 + 8 q}}^{2} = {(q + 3)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{24 + 8 q}$ as ${(24 + 8 q)}^{\frac{1}{2}}$ .

${({(24 + 8 q)}^{\frac{1}{2}})}^{2} = {(q + 3)}^{2}$

Simplify the left side.

Simplify ${({(24 + 8 q)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(24 + 8 q)}^{\frac{1}{2}})}^{2}$ .

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

${(24 + 8 q)}^{\frac{1}{2} \cdot 2} = {(q + 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(24 + 8 q)}^{\frac{1}{2} \cdot 2} = {(q + 3)}^{2}$

Rewrite the expression.

${(24 + 8 q)}^{1} = {(q + 3)}^{2}$

Simplify.

$24 + 8 q = {(q + 3)}^{2}$

Simplify the right side.

Simplify ${(q + 3)}^{2}$ .

Rewrite ${(q + 3)}^{2}$ as $(q + 3) (q + 3)$ .

$24 + 8 q = (q + 3) (q + 3)$

Expand $(q + 3) (q + 3)$ using the FOIL Method.

Apply the distributive property.

$24 + 8 q = q (q + 3) + 3 (q + 3)$

Apply the distributive property.

$24 + 8 q = q \cdot q + q \cdot 3 + 3 (q + 3)$

Apply the distributive property.

$24 + 8 q = q \cdot q + q \cdot 3 + 3 q + 3 \cdot 3$

Simplify and combine like terms.

Simplify each term.

Multiply $q$ by $q$ .

$24 + 8 q = q^{2} + q \cdot 3 + 3 q + 3 \cdot 3$

Move $3$ to the left of $q$ .

$24 + 8 q = q^{2} + 3 \cdot q + 3 q + 3 \cdot 3$

Multiply $3$ by $3$ .

$24 + 8 q = q^{2} + 3 q + 3 q + 9$

Add $3 q$ and $3 q$ .

$24 + 8 q = q^{2} + 6 q + 9$

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.

$q^{2} + 6 q + 9 = 24 + 8 q$

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

Subtract $8 q$ from both sides of the equation.

$q^{2} + 6 q + 9 - 8 q = 24$

Subtract $8 q$ from $6 q$ .

$q^{2} - 2 q + 9 = 24$

Subtract $24$ from both sides of the equation.

$q^{2} - 2 q + 9 - 24 = 0$

Subtract $24$ from $9$ .

$q^{2} - 2 q - 15 = 0$

Factor $q^{2} - 2 q - 15$ 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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Write the factored form using these integers.

$(q - 5) (q + 3) = 0$

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

$q - 5 = 0$

$q + 3 = 0$

Set $q - 5$ equal to $0$ and solve for $q$ .

Set $q - 5$ equal to $0$ .

$q - 5 = 0$

Add $5$ to both sides of the equation.

$q = 5$

Set $q + 3$ equal to $0$ and solve for $q$ .

Set $q + 3$ equal to $0$ .

$q + 3 = 0$

Subtract $3$ from both sides of the equation.

$q = - 3$

The final solution is all the values that make $(q - 5) (q + 3) = 0$ true.

$q = 5, - 3$

Solve for q square root of 24+8q=q+3

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