Calculus Examples

$y = 9 x - x^{2}$

Step 1

Rewrite the equation as $9 x - x^{2} = y$ .

$9 x - x^{2} = y$

Step 2

Subtract $y$ from both sides of the equation.

$9 x - x^{2} - y = 0$

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 $a = - 1$ , $b = 9$ , and $c = - y$ into the quadratic formula and solve for $x$ .

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot - 1 \cdot (- 1 y)}}{2 \cdot - 1}$

Step 5.1.2

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

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Step 5.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 9 \pm \sqrt{81 + 4 \cdot (- 1 y)}}{2 \cdot - 1}$

Step 5.1.2.2

Multiply $4$ by $- 1$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 y}}{2 \cdot - 1}$

Step 5.2

Multiply $2$ by $- 1$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 y}}{- 2}$

Step 5.3

Simplify $\frac{- 9 \pm \sqrt{81 - 4 y}}{- 2}$ .

$x = \frac{9 \pm \sqrt{81 - 4 y}}{2}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{9 + \sqrt{81 - 4 y}}{2}$

$x = \frac{9 - \sqrt{81 - 4 y}}{2}$