Precalculus Examples

$x = t^{2} + t$ , $y = \sqrt{t^{2} - t}$

Step 1

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

$x = t^{2} + t$

Step 2

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

$t^{2} + t = x$

Step 3

Subtract $x$ from both sides of the equation.

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

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

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

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

One to any power is one.

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

Step 6.1.2

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

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

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

Step 6.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 6.2

Multiply $2$ by $1$ .

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

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

One to any power is one.

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

Step 7.1.2

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

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

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

Step 7.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 7.2

Multiply $2$ by $1$ .

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

Step 7.3

Change the $\pm$ to $+$ .

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

Step 7.4

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 7.5

Factor $- 1$ out of $\sqrt{1 + 4 x}$ .

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

Step 7.6

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{1 + 4 x})$ .

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

Step 7.7

Move the negative in front of the fraction.

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

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

One to any power is one.

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

Step 8.1.2

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

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

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

Step 8.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 8.2

Multiply $2$ by $1$ .

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

Step 8.3

Change the $\pm$ to $-$ .

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

Step 8.4

Factor $- 1$ out of $- 1 - \sqrt{1 + 4 x}$ .

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Step 8.4.1

Reorder $1$ and $4 x$ .

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

Step 8.4.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 8.4.3

Factor $- 1$ out of $- \sqrt{4 x + 1}$ .

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

Step 8.4.4

Factor $- 1$ out of $- 1 (1) - (\sqrt{4 x + 1})$ .

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

Step 8.4.5

Rewrite $- 1 (1 + \sqrt{4 x + 1})$ as $- (1 + \sqrt{4 x + 1})$ .

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

Step 8.5

Move the negative in front of the fraction.

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

Step 9

The final answer is the combination of both solutions.

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

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

Step 10

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

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

Step 11

Multiply $- 1$ by $- \frac{1 - \sqrt{1 + 4 x}}{2}, - \frac{1 + \sqrt{4 x + 1}}{2}$ .

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

Step 12

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

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Step 12.1

To write ${(- \frac{1 - \sqrt{1 + 4 x}}{2}, - \frac{1 + \sqrt{4 x + 1}}{2})}^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 12.2

Simplify terms.

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Step 12.2.1

Combine ${(- \frac{1 - \sqrt{1 + 4 x}}{2}, - \frac{1 + \sqrt{4 x + 1}}{2})}^{2}$ and $\frac{2}{2}$ .

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

Step 12.2.2

Combine the numerators over the common denominator.

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

Step 12.3

Move $2$ to the left of ${(- \frac{1 - \sqrt{1 + 4 x}}{2}, - \frac{1 + \sqrt{4 x + 1}}{2})}^{2}$ .

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