Precalculus Examples

$y > 4 x^{2} - 12 x$

Step 1

Rewrite so $x$ is on the left side of the inequality.

$4 x^{2} - 12 x < y$

Step 2

Subtract $y$ from both sides of the inequality.

$4 x^{2} - 12 x - y < 0$

Step 3

Convert the inequality to an equation.

$4 x^{2} - 12 x - y = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 4$ , $b = - 12$ , and $c = - y$ into the quadratic formula and solve for $x$ .

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

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 4 \cdot (- 1 y)}}{2 \cdot 4}$

Step 6.1.2

Multiply $- 4 \cdot 4 \cdot - 1$ .

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Step 6.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{12 \pm \sqrt{144 - 16 \cdot (- 1 y)}}{2 \cdot 4}$

Step 6.1.2.2

Multiply $- 16$ by $- 1$ .

$x = \frac{12 \pm \sqrt{144 + 16 y}}{2 \cdot 4}$

Step 6.1.3

Factor $16$ out of $144 + 16 y$ .

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Step 6.1.3.1

Factor $16$ out of $144$ .

$x = \frac{12 \pm \sqrt{16 \cdot 9 + 16 y}}{2 \cdot 4}$

Step 6.1.3.2

Factor $16$ out of $16 \cdot 9 + 16 y$ .

$x = \frac{12 \pm \sqrt{16 (9 + y)}}{2 \cdot 4}$

Step 6.1.4

Rewrite $16 (9 + y)$ as ${(2^{2})}^{2} (9 + y)$ .

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Step 6.1.4.1

Rewrite $16$ as $4^{2}$ .

$x = \frac{12 \pm \sqrt{4^{2} (9 + y)}}{2 \cdot 4}$

Step 6.1.4.2

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

$x = \frac{12 \pm \sqrt{{(2^{2})}^{2} (9 + y)}}{2 \cdot 4}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{12 \pm 2^{2} \sqrt{9 + y}}{2 \cdot 4}$

Step 6.1.6

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

$x = \frac{12 \pm 4 \sqrt{9 + y}}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$x = \frac{12 \pm 4 \sqrt{9 + y}}{8}$

Step 6.3

Simplify $\frac{12 \pm 4 \sqrt{9 + y}}{8}$ .

$x = \frac{3 \pm \sqrt{9 + y}}{2}$

Step 7

Consolidate the solutions.

$x = \frac{3 + \sqrt{9 + y}}{2}$

$x = \frac{3 - \sqrt{9 + y}}{2}$