Solve for a square root of 12-a+7a=8a

$\sqrt{12 - a} + 7 a = 8 a$

Step 1

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

Subtract $7 a$ from both sides of the equation.

$\sqrt{12 - a} = 8 a - 7 a$

Subtract $7 a$ from $8 a$ .

$\sqrt{12 - a} = a$

Step 2

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

${\sqrt{12 - a}}^{2} = a^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{12 - a}$ as ${(12 - a)}^{\frac{1}{2}}$ .

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

Simplify the left side.

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

Multiply the exponents in ${({(12 - a)}^{\frac{1}{2}})}^{2}$ .

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

${(12 - a)}^{\frac{1}{2} \cdot 2} = a^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(12 - a)}^{\frac{1}{2} \cdot 2} = a^{2}$

Rewrite the expression.

${(12 - a)}^{1} = a^{2}$

Simplify.

$12 - a = a^{2}$

Step 4

Solve for $a$ .

Subtract $a^{2}$ from both sides of the equation.

$12 - a - a^{2} = 0$

Factor the left side of the equation.

Factor $- 1$ out of $12 - a - a^{2}$ .

Reorder the expression.

Move $12$ .

$- a - a^{2} + 12 = 0$

Reorder $- a$ and $- a^{2}$ .

$- a^{2} - a + 12 = 0$

Factor $- 1$ out of $- a^{2}$ .

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

Factor $- 1$ out of $- a$ .

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

Rewrite $12$ as $- 1 (- 12)$ .

$- (a^{2}) - (a) - 1 \cdot - 12 = 0$

Factor $- 1$ out of $- (a^{2}) - (a)$ .

$- (a^{2} + a) - 1 \cdot - 12 = 0$

Factor $- 1$ out of $- (a^{2} + a) - 1 (- 12)$ .

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

Factor.

Factor $a^{2} + a - 12$ 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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Write the factored form using these integers.

$- ((a - 3) (a + 4)) = 0$

Remove unnecessary parentheses.

$- (a - 3) (a + 4) = 0$

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

$a - 3 = 0$

$a + 4 = 0$

Set $a - 3$ equal to $0$ and solve for $a$ .

Set $a - 3$ equal to $0$ .

$a - 3 = 0$

Add $3$ to both sides of the equation.

$a = 3$

Set $a + 4$ equal to $0$ and solve for $a$ .

Set $a + 4$ equal to $0$ .

$a + 4 = 0$

Subtract $4$ from both sides of the equation.

$a = - 4$

The final solution is all the values that make $- (a - 3) (a + 4) = 0$ true.

$a = 3, - 4$

Step 5

Exclude the solutions that do not make $\sqrt{12 - a} + 7 a = 8 a$ true.

$a = 3$

Solve for a square root of 12-a+7a=8a

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