Calculus Examples

y = 4 x - x^{2}

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

Step 1

Rewrite the equation as

4 x - x^{2} = y

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

4 x - x^{2} = y

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

Step 2

Subtract

y

$y$ from both sides of the equation.

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

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

Step 3

Use the quadratic formula to find the solutions.

\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values

a = - 1

$a = - 1$ ,

b = 4

$b = 4$ , and

c = - y

$c = - y$ into the quadratic formula and solve for

x

$x$ .

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

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise

4

$4$ to the power of

2

$2$ .

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

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

Step 5.1.2

Multiply

- 4 \cdot - 1 \cdot - 1

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

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Step 5.1.2.1

Multiply

- 4

$- 4$ by

- 1

$- 1$ .

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

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

Step 5.1.2.2

Multiply

4

$4$ by

- 1

$- 1$ .

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

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

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

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

Step 5.1.3

Factor

4

$4$ out of

16 - 4 y

$16 - 4 y$ .

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Step 5.1.3.1

Factor

4

$4$ out of

16

$16$ .

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

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

Step 5.1.3.2

Factor

4

$4$ out of

- 4 y

$- 4 y$ .

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

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

Step 5.1.3.3

Factor

4

$4$ out of

4 (4) + 4 (- y)

$4 (4) + 4 (- y)$ .

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

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

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

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

Step 5.1.4

Rewrite

4 (4 - y)

$4 (4 - y)$ as

2^{2} (2^{2} - y)

$2^{2} (2^{2} - y)$ .

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Step 5.1.4.1

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

Step 5.1.4.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

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

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

Step 5.1.5

Pull terms out from under the radical.

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

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

Step 5.1.6

Raise

2

$2$ to the power of

2

$2$ .

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

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

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

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

Step 5.2

Multiply

2

$2$ by

- 1

$- 1$ .

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

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

Step 5.3

Simplify

\frac{- 4 \pm 2 \sqrt{4 - y}}{- 2}

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

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

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

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

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

Step 6

The final answer is the combination of both solutions.

x = 2 + \sqrt{4 - y}

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

x = 2 - \sqrt{4 - y}

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