Precalculus Examples

$x = t^{2} + t + 1$ , $y = 2 t$

Step 1

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

$x = t^{2} + t + 1$

Step 2

Rewrite the equation as $t^{2} + t + 1 = x$ .

$t^{2} + t + 1 = x$

Step 3

Subtract $x$ from both sides of the equation.

$t^{2} + t + 1 - x = 0$

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

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

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 $- 4$ by $1$ .

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

Step 6.1.3

Apply the distributive property.

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

Step 6.1.4

Multiply $- 4$ by $1$ .

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

Step 6.1.5

Multiply $- 1$ by $- 4$ .

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

Step 6.1.6

Subtract $4$ from $1$ .

$t = \frac{- 1 \pm \sqrt{- 3 + 4 x}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$t = \frac{- 1 \pm \sqrt{- 3 + 4 x}}{2}$

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 $- 4$ by $1$ .

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

Step 7.1.3

Apply the distributive property.

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

Step 7.1.4

Multiply $- 4$ by $1$ .

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

Step 7.1.5

Multiply $- 1$ by $- 4$ .

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

Step 7.1.6

Subtract $4$ from $1$ .

$t = \frac{- 1 \pm \sqrt{- 3 + 4 x}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$t = \frac{- 1 \pm \sqrt{- 3 + 4 x}}{2}$

Step 7.3

Change the $\pm$ to $+$ .

$t = \frac{- 1 + \sqrt{- 3 + 4 x}}{2}$

Step 7.4

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

$t = \frac{- 1 \cdot 1 + \sqrt{- 3 + 4 x}}{2}$

Step 7.5

Factor $- 1$ out of $\sqrt{- 3 + 4 x}$ .

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

Step 7.6

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{- 3 + 4 x})$ .

$t = \frac{- 1 (1 - \sqrt{- 3 + 4 x})}{2}$

Step 7.7

Move the negative in front of the fraction.

$t = - \frac{1 - \sqrt{- 3 + 4 x}}{2}$

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 $- 4$ by $1$ .

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

Step 8.1.3

Apply the distributive property.

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

Step 8.1.4

Multiply $- 4$ by $1$ .

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

Step 8.1.5

Multiply $- 1$ by $- 4$ .

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

Step 8.1.6

Subtract $4$ from $1$ .

$t = \frac{- 1 \pm \sqrt{- 3 + 4 x}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$t = \frac{- 1 \pm \sqrt{- 3 + 4 x}}{2}$

Step 8.3

Change the $\pm$ to $-$ .

$t = \frac{- 1 - \sqrt{- 3 + 4 x}}{2}$

Step 8.4

Factor $- 1$ out of $- 1 - \sqrt{- 3 + 4 x}$ .

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Step 8.4.1

Reorder $- 3$ and $4 x$ .

$t = \frac{- 1 - \sqrt{4 x - 3}}{2}$

Step 8.4.2

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

$t = \frac{- 1 \cdot 1 - \sqrt{4 x - 3}}{2}$

Step 8.4.3

Factor $- 1$ out of $- \sqrt{4 x - 3}$ .

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

Step 8.4.4

Factor $- 1$ out of $- 1 (1) - (\sqrt{4 x - 3})$ .

$t = \frac{- 1 (1 + \sqrt{4 x - 3})}{2}$

Step 8.4.5

Rewrite $- 1 (1 + \sqrt{4 x - 3})$ as $- (1 + \sqrt{4 x - 3})$ .

$t = \frac{- (1 + \sqrt{4 x - 3})}{2}$

Step 8.5

Move the negative in front of the fraction.

$t = - \frac{1 + \sqrt{4 x - 3}}{2}$

Step 9

The final answer is the combination of both solutions.

$t = - \frac{1 - \sqrt{- 3 + 4 x}}{2}$

$t = - \frac{1 + \sqrt{4 x - 3}}{2}$

Step 10

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

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

Step 11

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

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Step 11.1

Multiply $2$ by each element of the matrix.

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

Step 11.2

Simplify each element in the matrix.

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Step 11.2.1

Cancel the common factor of $2$ .

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Step 11.2.1.1

Move the leading negative in $- \frac{1 - \sqrt{- 3 + 4 x}}{2}$ into the numerator.

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

Step 11.2.1.2

Cancel the common factor.

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

Step 11.2.1.3

Rewrite the expression.

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

Step 11.2.2

Apply the distributive property.

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

Step 11.2.3

Multiply $- 1$ by $1$ .

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

Step 11.2.4

Multiply $- - \sqrt{- 3 + 4 x}$ .

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Step 11.2.4.1

Multiply $- 1$ by $- 1$ .

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

Step 11.2.4.2

Multiply $\sqrt{- 3 + 4 x}$ by $1$ .

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

Step 11.2.5

Cancel the common factor of $2$ .

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Step 11.2.5.1

Move the leading negative in $- \frac{1 + \sqrt{4 x - 3}}{2}$ into the numerator.

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

Step 11.2.5.2

Cancel the common factor.

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

Step 11.2.5.3

Rewrite the expression.

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

Step 11.2.6

Apply the distributive property.

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

Step 11.2.7

Multiply $- 1$ by $1$ .

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