Solve for a y=3a^2-12a+8

$y = 3 a^{2} - 12 a + 8$

Step 1

Rewrite the equation as $3 a^{2} - 12 a + 8 = y$ .

$3 a^{2} - 12 a + 8 = y$

Step 2

Subtract $y$ from both sides of the equation.

$3 a^{2} - 12 a + 8 - y = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 3$ , $b = - 12$ , and $c = 8 - y$ into the quadratic formula and solve for $a$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (3 \cdot (8 - y))}}{2 \cdot 3}$

Step 5

Simplify.

Simplify the numerator.

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

$a = \frac{12 \pm \sqrt{144 - 4 \cdot 3 \cdot (8 - y)}}{2 \cdot 3}$

Multiply $- 4$ by $3$ .

$a = \frac{12 \pm \sqrt{144 - 12 \cdot (8 - y)}}{2 \cdot 3}$

Apply the distributive property.

$a = \frac{12 \pm \sqrt{144 - 12 \cdot 8 - 12 (- y)}}{2 \cdot 3}$

Multiply $- 12$ by $8$ .

$a = \frac{12 \pm \sqrt{144 - 96 - 12 (- y)}}{2 \cdot 3}$

Multiply $- 1$ by $- 12$ .

$a = \frac{12 \pm \sqrt{144 - 96 + 12 y}}{2 \cdot 3}$

Subtract $96$ from $144$ .

$a = \frac{12 \pm \sqrt{48 + 12 y}}{2 \cdot 3}$

Factor $12$ out of $48 + 12 y$ .

Factor $12$ out of $48$ .

$a = \frac{12 \pm \sqrt{12 \cdot 4 + 12 y}}{2 \cdot 3}$

Factor $12$ out of $12 \cdot 4 + 12 y$ .

$a = \frac{12 \pm \sqrt{12 (4 + y)}}{2 \cdot 3}$

Rewrite $12 (4 + y)$ as $2^{2} \cdot (3 (2^{2} + y))$ .

Factor $4$ out of $12$ .

$a = \frac{12 \pm \sqrt{4 (3) (4 + y)}}{2 \cdot 3}$

Rewrite $4$ as $2^{2}$ .

$a = \frac{12 \pm \sqrt{2^{2} \cdot (3 (4 + y))}}{2 \cdot 3}$

Rewrite $4$ as $2^{2}$ .

$a = \frac{12 \pm \sqrt{2^{2} \cdot (3 (2^{2} + y))}}{2 \cdot 3}$

Add parentheses.

$a = \frac{12 \pm \sqrt{2^{2} \cdot (3 (2^{2} + y))}}{2 \cdot 3}$

Pull terms out from under the radical.

$a = \frac{12 \pm 2 \sqrt{3 (2^{2} + y)}}{2 \cdot 3}$

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

$a = \frac{12 \pm 2 \sqrt{3 (4 + y)}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$a = \frac{12 \pm 2 \sqrt{3 (4 + y)}}{6}$

Simplify $\frac{12 \pm 2 \sqrt{3 (4 + y)}}{6}$ .

$a = \frac{6 \pm \sqrt{3 (4 + y)}}{3}$

Step 6

The final answer is the combination of both solutions.

$a = \frac{6 + \sqrt{3 (4 + y)}}{3}$

$a = \frac{6 - \sqrt{3 (4 + y)}}{3}$

Solve for a y=3a^2-12a+8

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