Finite Math Examples

$y > x^{2} + 4 x$

Step 1

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

$x^{2} + 4 x < y$

Step 2

Subtract $y$ from both sides of the inequality.

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

Step 3

Convert the inequality to an equation.

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

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 $a = 1$ , $b = 4$ , and $c = - y$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot (- y))}}{2 \cdot 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $4$ to the power of $2$ .

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

Step 6.1.2

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

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

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

Step 6.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{- 4 \pm \sqrt{16 + 4 y}}{2 \cdot 1}$

Step 6.1.3

Factor $4$ out of $16 + 4 y$ .

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Step 6.1.3.1

Factor $4$ out of $16$ .

$x = \frac{- 4 \pm \sqrt{4 \cdot 4 + 4 y}}{2 \cdot 1}$

Step 6.1.3.2

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

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

Step 6.1.4

Rewrite $4 (4 + y)$ as $2^{2} (2^{2} + y)$ .

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Step 6.1.4.1

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

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

Step 6.1.4.2

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

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

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{2^{2} + y}}{2 \cdot 1}$

Step 6.1.6

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

$x = \frac{- 4 \pm 2 \sqrt{4 + y}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 \sqrt{4 + y}}{2}$

Step 6.3

Simplify $\frac{- 4 \pm 2 \sqrt{4 + y}}{2}$ .

$x = - 2 \pm \sqrt{4 + y}$

Step 7

Consolidate the solutions.

$x = - 2 + \sqrt{4 + y}$

$x = - 2 - \sqrt{4 + y}$