Trigonometry Examples

$y = 2 x^{2} - 4 x - 1$

Step 1

Rewrite the equation as $2 x^{2} - 4 x - 1 = y$ .

$2 x^{2} - 4 x - 1 = y$

Step 2

Subtract $y$ from both sides of the equation.

$2 x^{2} - 4 x - 1 - y = 0$

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 $a = 2$ , $b = - 4$ , and $c = - 1 - y$ into the quadratic formula and solve for $x$ .

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 2 \cdot (- 1 - y)$ .

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Step 5.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

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

Step 5.1.1.2

Factor $- 4$ out of $- 4 \cdot 2 \cdot (- 1 - y)$ .

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

Step 5.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (2 \cdot (- 1 - y))$ .

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

Step 5.1.2

Factor $2$ out of $- 4 + 2 \cdot (- 1 - y)$ .

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Step 5.1.2.1

Factor $2$ out of $- 4$ .

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

Step 5.1.2.2

Factor $2$ out of $2 \cdot - 2 + 2 \cdot (- 1 - y)$ .

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

Step 5.1.3

Subtract $1$ from $- 2$ .

$x = \frac{4 \pm \sqrt{- 4 \cdot (2 (- 3 - y))}}{2 \cdot 2}$

Step 5.1.4

Multiply $- 4$ by $2$ .

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

Step 5.1.5

Rewrite $- 8 (- 3 - y)$ as $2^{2} \cdot (- 2 (- 3 - y))$ .

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Step 5.1.5.1

Factor $4$ out of $- 8$ .

$x = \frac{4 \pm \sqrt{4 (- 2) (- 3 - y)}}{2 \cdot 2}$

Step 5.1.5.2

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

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

Step 5.1.5.3

Add parentheses.

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

Step 5.1.6

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{- 2 (- 3 - y)}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$x = \frac{4 \pm 2 \sqrt{- 2 (- 3 - y)}}{4}$

Step 5.3

Simplify $\frac{4 \pm 2 \sqrt{- 2 (- 3 - y)}}{4}$ .

$x = \frac{2 \pm \sqrt{- 2 (- 3 - y)}}{2}$

Step 6

The final answer is the combination of both solutions.

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

$x = \frac{2 - \sqrt{- 2 (- 3 - y)}}{2}$