Calculus Examples

$x (t) = t^{2} - 4 t$ and $y (t) = 2 t^{3} - 6 t$

Step 1

Set up the parametric equation for $x (t)$ to solve the equation for $t$ .

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

Step 2

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

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

Step 3

Subtract $x$ from both sides of the equation.

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

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

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

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

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Step 6.1.1.1

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

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

Step 6.1.1.2

Factor $- 4$ out of $- 4 \cdot 1 \cdot - 1 x$ .

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

Step 6.1.1.3

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

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

Step 6.1.2

Multiply $- 1$ by $1$ .

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

Step 6.1.3

Rewrite $- 4 (- 4 - x)$ as $2^{2} \cdot (- 1 (- 4 - x))$ .

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Step 6.1.3.1

Factor $4$ out of $- 4$ .

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

Step 6.1.3.2

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

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

Step 6.1.3.3

Add parentheses.

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

Step 6.1.4

Pull terms out from under the radical.

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

Step 6.2

Multiply $2$ by $1$ .

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

Step 6.3

Simplify $\frac{4 \pm 2 \sqrt{- 1 (- 4 - x)}}{2}$ .

$t = 2 \pm \sqrt{- 1 (- 4 - x)}$

Step 7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

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Step 7.1.1.1

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

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

Step 7.1.1.2

Factor $- 4$ out of $- 4 \cdot 1 \cdot - 1 x$ .

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

Step 7.1.1.3

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

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

Step 7.1.2

Multiply $- 1$ by $1$ .

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

Step 7.1.3

Rewrite $- 4 (- 4 - x)$ as $2^{2} \cdot (- 1 (- 4 - x))$ .

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Step 7.1.3.1

Factor $4$ out of $- 4$ .

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

Step 7.1.3.2

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

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

Step 7.1.3.3

Add parentheses.

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

Step 7.1.4

Pull terms out from under the radical.

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

Step 7.2

Multiply $2$ by $1$ .

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

Step 7.3

Simplify $\frac{4 \pm 2 \sqrt{- 1 (- 4 - x)}}{2}$ .

$t = 2 \pm \sqrt{- 1 (- 4 - x)}$

Step 7.4

Change the $\pm$ to $+$ .

$t = 2 + \sqrt{- 1 (- 4 - x)}$

Step 8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

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

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Step 8.1.1.1

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

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

Step 8.1.1.2

Factor $- 4$ out of $- 4 \cdot 1 \cdot - 1 x$ .

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

Step 8.1.1.3

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

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

Step 8.1.2

Multiply $- 1$ by $1$ .

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

Step 8.1.3

Rewrite $- 4 (- 4 - x)$ as $2^{2} \cdot (- 1 (- 4 - x))$ .

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Step 8.1.3.1

Factor $4$ out of $- 4$ .

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

Step 8.1.3.2

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

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

Step 8.1.3.3

Add parentheses.

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

Step 8.1.4

Pull terms out from under the radical.

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

Step 8.2

Multiply $2$ by $1$ .

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

Step 8.3

Simplify $\frac{4 \pm 2 \sqrt{- 1 (- 4 - x)}}{2}$ .

$t = 2 \pm \sqrt{- 1 (- 4 - x)}$

Step 8.4

Change the $\pm$ to $-$ .

$t = 2 - \sqrt{- 1 (- 4 - x)}$

Step 9

The final answer is the combination of both solutions.

$t = 2 + \sqrt{- 1 (- 4 - x)}$

$t = 2 - \sqrt{- 1 (- 4 - x)}$

Step 10

Replace $t$ in the equation for $y$ to get the equation in terms of $x$ .

$y = 2 {(2 + \sqrt{- 1 (- 4 - x)}, 2 - \sqrt{- 1 (- 4 - x)})}^{3} - 6 (2 + \sqrt{- 1 (- 4 - x)}, 2 - \sqrt{- 1 (- 4 - x)})$

Step 11

Simplify $2 {(2 + \sqrt{- 1 (- 4 - x)}, 2 - \sqrt{- 1 (- 4 - x)})}^{3} - 6 (2 + \sqrt{- 1 (- 4 - x)}, 2 - \sqrt{- 1 (- 4 - x)})$ .

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Step 11.1

Simplify each term.

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Step 11.1.1

Multiply $- 6$ by each element of the matrix.

$y = 2 {(2 + \sqrt{- 1 (- 4 - x)}, 2 - \sqrt{- 1 (- 4 - x)})}^{3} + (- 6 (2 + \sqrt{- 1 (- 4 - x)}), - 6 (2 - \sqrt{- 1 (- 4 - x)}))$

Step 11.1.2

Simplify each element in the matrix.

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Step 11.1.2.1

Apply the distributive property.

$y = 2 {(2 + \sqrt{- 1 (- 4 - x)}, 2 - \sqrt{- 1 (- 4 - x)})}^{3} + (- 6 \cdot 2 - 6 \sqrt{- 1 (- 4 - x)}, - 6 (2 - \sqrt{- 1 (- 4 - x)}))$

Step 11.1.2.2

Multiply $- 6$ by $2$ .

$y = 2 {(2 + \sqrt{- 1 (- 4 - x)}, 2 - \sqrt{- 1 (- 4 - x)})}^{3} + (- 12 - 6 \sqrt{- 1 (- 4 - x)}, - 6 (2 - \sqrt{- 1 (- 4 - x)}))$

Step 11.1.2.3

Apply the distributive property.

$y = 2 {(2 + \sqrt{- 1 (- 4 - x)}, 2 - \sqrt{- 1 (- 4 - x)})}^{3} + (- 12 - 6 \sqrt{- 1 (- 4 - x)}, - 6 \cdot 2 - 6 (- \sqrt{- 1 (- 4 - x)}))$

Step 11.1.2.4

Multiply $- 6$ by $2$ .

$y = 2 {(2 + \sqrt{- 1 (- 4 - x)}, 2 - \sqrt{- 1 (- 4 - x)})}^{3} + (- 12 - 6 \sqrt{- 1 (- 4 - x)}, - 12 - 6 (- \sqrt{- 1 (- 4 - x)}))$

Step 11.1.2.5

Multiply $- 1$ by $- 6$ .

$y = 2 {(2 + \sqrt{- 1 (- 4 - x)}, 2 - \sqrt{- 1 (- 4 - x)})}^{3} + (- 12 - 6 \sqrt{- 1 (- 4 - x)}, - 12 + 6 \sqrt{- 1 (- 4 - x)})$

Step 11.2

Simplify with commuting.

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Step 11.2.1

Reorder $- 4$ and $- x$ .

$y = 2 {(2 + \sqrt{- 1 (- x - 4)}, 2 - \sqrt{- 1 (- 4 - x)})}^{3} + (- 12 - 6 \sqrt{- 1 (- 4 - x)}, - 12 + 6 \sqrt{- 1 (- 4 - x)})$

Step 11.2.2

Reorder $- 4$ and $- x$ .

$y = 2 {(2 + \sqrt{- 1 (- x - 4)}, 2 - \sqrt{- 1 (- x - 4)})}^{3} + (- 12 - 6 \sqrt{- 1 (- 4 - x)}, - 12 + 6 \sqrt{- 1 (- 4 - x)})$

Step 11.2.3

Reorder $- 4$ and $- x$ .

$y = 2 {(2 + \sqrt{- 1 (- x - 4)}, 2 - \sqrt{- 1 (- x - 4)})}^{3} + (- 12 - 6 \sqrt{- 1 (- x - 4)}, - 12 + 6 \sqrt{- 1 (- 4 - x)})$

Step 11.2.4

Reorder $- 4$ and $- x$ .

$y = 2 {(2 + \sqrt{- 1 (- x - 4)}, 2 - \sqrt{- 1 (- x - 4)})}^{3} + (- 12 - 6 \sqrt{- 1 (- x - 4)}, - 12 + 6 \sqrt{- 1 (- x - 4)})$