Algebra Examples

$y < 2 x^{2} + 5 x - 1$

Step 1

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

$2 x^{2} + 5 x - 1 > y$

Step 2

Subtract $y$ from both sides of the inequality.

$2 x^{2} + 5 x - 1 - y > 0$

Step 3

Convert the inequality to an equation.

$2 x^{2} + 5 x - 1 - y = 0$

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 $a = 2$ , $b = 5$ , and $c = - 1 - y$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (2 \cdot (- 1 - y))}}{2 \cdot 2}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 2 \cdot (- 1 - y)}}{2 \cdot 2}$

Step 6.1.2

Multiply $- 4$ by $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 8 \cdot (- 1 - y)}}{2 \cdot 2}$

Step 6.1.3

Apply the distributive property.

$x = \frac{- 5 \pm \sqrt{25 - 8 \cdot - 1 - 8 (- y)}}{2 \cdot 2}$

Step 6.1.4

Multiply $- 8$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{25 + 8 - 8 (- y)}}{2 \cdot 2}$

Step 6.1.5

Multiply $- 1$ by $- 8$ .

$x = \frac{- 5 \pm \sqrt{25 + 8 + 8 y}}{2 \cdot 2}$

Step 6.1.6

Add $25$ and $8$ .

$x = \frac{- 5 \pm \sqrt{33 + 8 y}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$x = \frac{- 5 \pm \sqrt{33 + 8 y}}{4}$

Step 7

Consolidate the solutions.

$x = - \frac{5 - \sqrt{33 + 8 y}}{4}$

$x = - \frac{5 + \sqrt{8 y + 33}}{4}$