Trigonometry Examples

$y = - x^{2} - 3 x - 1$

Step 1

Rewrite the equation as $- x^{2} - 3 x - 1 = y$ .

$- x^{2} - 3 x - 1 = y$

Step 2

Subtract $y$ from both sides of the equation.

$- x^{2} - 3 x - 1 - y = 0$

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 $a = - 1$ , $b = - 3$ , and $c = - 1 - y$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (- 1 \cdot (- 1 - y))}}{2 \cdot - 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 1 \cdot (- 1 - y)}}{2 \cdot - 1}$

Step 5.1.2

Multiply $- 4$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 4 \cdot (- 1 - y)}}{2 \cdot - 1}$

Step 5.1.3

Apply the distributive property.

$x = \frac{3 \pm \sqrt{9 + 4 \cdot - 1 + 4 (- y)}}{2 \cdot - 1}$

Step 5.1.4

Multiply $4$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 - 4 + 4 (- y)}}{2 \cdot - 1}$

Step 5.1.5

Multiply $- 1$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 4 - 4 y}}{2 \cdot - 1}$

Step 5.1.6

Subtract $4$ from $9$ .

$x = \frac{3 \pm \sqrt{5 - 4 y}}{2 \cdot - 1}$

Step 5.2

Multiply $2$ by $- 1$ .

$x = \frac{3 \pm \sqrt{5 - 4 y}}{- 2}$

Step 5.3

Move the negative in front of the fraction.

$x = - \frac{3 \pm \sqrt{5 - 4 y}}{2}$

Step 6

The final answer is the combination of both solutions.

$x = - \frac{3 + \sqrt{5 - 4 y}}{2}$

$x = - \frac{3 - \sqrt{5 - 4 y}}{2}$