Precalculus Examples

$5 x^{2} + 10 x + 5 y^{2} + 19 y = 9$

Step 1

Subtract $9$ from both sides of the equation.

$5 x^{2} + 10 x + 5 y^{2} + 19 y - 9 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 5$ , $b = 19$ , and $c = 5 x^{2} + 10 x - 9$ into the quadratic formula and solve for $y$ .

$\frac{- 19 \pm \sqrt{19^{2} - 4 \cdot (5 \cdot (5 x^{2} + 10 x - 9))}}{2 \cdot 5}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $19$ to the power of $2$ .

$y = \frac{- 19 \pm \sqrt{361 - 4 \cdot 5 \cdot (5 x^{2} + 10 x - 9)}}{2 \cdot 5}$

Step 4.1.2

Multiply $- 4$ by $5$ .

$y = \frac{- 19 \pm \sqrt{361 - 20 \cdot (5 x^{2} + 10 x - 9)}}{2 \cdot 5}$

Step 4.1.3

Apply the distributive property.

$y = \frac{- 19 \pm \sqrt{361 - 20 (5 x^{2}) - 20 (10 x) - 20 \cdot - 9}}{2 \cdot 5}$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Multiply $5$ by $- 20$ .

$y = \frac{- 19 \pm \sqrt{361 - 100 x^{2} - 20 (10 x) - 20 \cdot - 9}}{2 \cdot 5}$

Step 4.1.4.2

Multiply $10$ by $- 20$ .

$y = \frac{- 19 \pm \sqrt{361 - 100 x^{2} - 200 x - 20 \cdot - 9}}{2 \cdot 5}$

Step 4.1.4.3

Multiply $- 20$ by $- 9$ .

$y = \frac{- 19 \pm \sqrt{361 - 100 x^{2} - 200 x + 180}}{2 \cdot 5}$

Step 4.1.5

Add $361$ and $180$ .

$y = \frac{- 19 \pm \sqrt{- 100 x^{2} - 200 x + 541}}{2 \cdot 5}$

Step 4.2

Multiply $2$ by $5$ .

$y = \frac{- 19 \pm \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $19$ to the power of $2$ .

$y = \frac{- 19 \pm \sqrt{361 - 4 \cdot 5 \cdot (5 x^{2} + 10 x - 9)}}{2 \cdot 5}$

Step 5.1.2

Multiply $- 4$ by $5$ .

$y = \frac{- 19 \pm \sqrt{361 - 20 \cdot (5 x^{2} + 10 x - 9)}}{2 \cdot 5}$

Step 5.1.3

Apply the distributive property.

$y = \frac{- 19 \pm \sqrt{361 - 20 (5 x^{2}) - 20 (10 x) - 20 \cdot - 9}}{2 \cdot 5}$

Step 5.1.4

Simplify.

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Step 5.1.4.1

Multiply $5$ by $- 20$ .

$y = \frac{- 19 \pm \sqrt{361 - 100 x^{2} - 20 (10 x) - 20 \cdot - 9}}{2 \cdot 5}$

Step 5.1.4.2

Multiply $10$ by $- 20$ .

$y = \frac{- 19 \pm \sqrt{361 - 100 x^{2} - 200 x - 20 \cdot - 9}}{2 \cdot 5}$

Step 5.1.4.3

Multiply $- 20$ by $- 9$ .

$y = \frac{- 19 \pm \sqrt{361 - 100 x^{2} - 200 x + 180}}{2 \cdot 5}$

Step 5.1.5

Add $361$ and $180$ .

$y = \frac{- 19 \pm \sqrt{- 100 x^{2} - 200 x + 541}}{2 \cdot 5}$

Step 5.2

Multiply $2$ by $5$ .

$y = \frac{- 19 \pm \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 5.3

Change the $\pm$ to $+$ .

$y = \frac{- 19 + \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 5.4

Rewrite $- 19$ as $- 1 (19)$ .

$y = \frac{- 1 \cdot 19 + \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 5.5

Factor $- 1$ out of $\sqrt{- 100 x^{2} - 200 x + 541}$ .

$y = \frac{- 1 \cdot 19 - 1 (- \sqrt{- 100 x^{2} - 200 x + 541})}{10}$

Step 5.6

Factor $- 1$ out of $- 1 (19) - 1 (- \sqrt{- 100 x^{2} - 200 x + 541})$ .

$y = \frac{- 1 (19 - \sqrt{- 100 x^{2} - 200 x + 541})}{10}$

Step 5.7

Move the negative in front of the fraction.

$y = - \frac{19 - \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $19$ to the power of $2$ .

$y = \frac{- 19 \pm \sqrt{361 - 4 \cdot 5 \cdot (5 x^{2} + 10 x - 9)}}{2 \cdot 5}$

Step 6.1.2

Multiply $- 4$ by $5$ .

$y = \frac{- 19 \pm \sqrt{361 - 20 \cdot (5 x^{2} + 10 x - 9)}}{2 \cdot 5}$

Step 6.1.3

Apply the distributive property.

$y = \frac{- 19 \pm \sqrt{361 - 20 (5 x^{2}) - 20 (10 x) - 20 \cdot - 9}}{2 \cdot 5}$

Step 6.1.4

Simplify.

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Step 6.1.4.1

Multiply $5$ by $- 20$ .

$y = \frac{- 19 \pm \sqrt{361 - 100 x^{2} - 20 (10 x) - 20 \cdot - 9}}{2 \cdot 5}$

Step 6.1.4.2

Multiply $10$ by $- 20$ .

$y = \frac{- 19 \pm \sqrt{361 - 100 x^{2} - 200 x - 20 \cdot - 9}}{2 \cdot 5}$

Step 6.1.4.3

Multiply $- 20$ by $- 9$ .

$y = \frac{- 19 \pm \sqrt{361 - 100 x^{2} - 200 x + 180}}{2 \cdot 5}$

Step 6.1.5

Add $361$ and $180$ .

$y = \frac{- 19 \pm \sqrt{- 100 x^{2} - 200 x + 541}}{2 \cdot 5}$

Step 6.2

Multiply $2$ by $5$ .

$y = \frac{- 19 \pm \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 6.3

Change the $\pm$ to $-$ .

$y = \frac{- 19 - \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 6.4

Factor $- 1$ out of $- 19 - \sqrt{- 100 x^{2} - 200 x + 541}$ .

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Step 6.4.1

Rewrite $- 19$ as $- 1 (19)$ .

$y = \frac{- 1 \cdot 19 - \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 6.4.2

Factor $- 1$ out of $- \sqrt{- 100 x^{2} - 200 x + 541}$ .

$y = \frac{- 1 \cdot 19 - (\sqrt{- 100 x^{2} - 200 x + 541})}{10}$

Step 6.4.3

Factor $- 1$ out of $- 1 (19) - (\sqrt{- 100 x^{2} - 200 x + 541})$ .

$y = \frac{- 1 (19 + \sqrt{- 100 x^{2} - 200 x + 541})}{10}$

Step 6.4.4

Rewrite $- 1 (19 + \sqrt{- 100 x^{2} - 200 x + 541})$ as $- (19 + \sqrt{- 100 x^{2} - 200 x + 541})$ .

$y = \frac{- (19 + \sqrt{- 100 x^{2} - 200 x + 541})}{10}$

Step 6.5

Move the negative in front of the fraction.

$y = - \frac{19 + \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 7

The final answer is the combination of both solutions.

$y = - \frac{19 - \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

$y = - \frac{19 + \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 8

The parent function is the simplest form of the type of function given.

$y = x^{2}$

Step 9

Solve $5 y^{2} + 19 y = 9 - 5 x^{2} - 10 x$ for $y$ .

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Step 9.1

Move all the expressions to the left side of the equation.

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Step 9.1.1

Subtract $9$ from both sides of the equation.

$5 y^{2} + 19 y - 9 = - 5 x^{2} - 10 x$

Step 9.1.2

Add $5 x^{2}$ to both sides of the equation.

$5 y^{2} + 19 y - 9 + 5 x^{2} = - 10 x$

Step 9.1.3

Add $10 x$ to both sides of the equation.

$5 y^{2} + 19 y - 9 + 5 x^{2} + 10 x = 0$

Step 9.2

Use the quadratic formula to find the solutions.

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

Step 9.3

Substitute the values $a = 5$ , $b = 19$ , and $c = - 9 + 5 x^{2} + 10 x$ into the quadratic formula and solve for $y$ .

$\frac{- 19 \pm \sqrt{19^{2} - 4 \cdot (5 \cdot (- 9 + 5 x^{2} + 10 x))}}{2 \cdot 5}$

Step 9.4

Simplify.

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Step 9.4.1

Simplify the numerator.

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Step 9.4.1.1

Raise $19$ to the power of $2$ .

$y = \frac{- 19 \pm \sqrt{361 - 4 \cdot 5 \cdot (- 9 + 5 x^{2} + 10 x)}}{2 \cdot 5}$

Step 9.4.1.2

Multiply $- 4$ by $5$ .

$y = \frac{- 19 \pm \sqrt{361 - 20 \cdot (- 9 + 5 x^{2} + 10 x)}}{2 \cdot 5}$

Step 9.4.1.3

Apply the distributive property.

$y = \frac{- 19 \pm \sqrt{361 - 20 \cdot - 9 - 20 (5 x^{2}) - 20 (10 x)}}{2 \cdot 5}$

Step 9.4.1.4

Simplify.

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Step 9.4.1.4.1

Multiply $- 20$ by $- 9$ .

$y = \frac{- 19 \pm \sqrt{361 + 180 - 20 (5 x^{2}) - 20 (10 x)}}{2 \cdot 5}$

Step 9.4.1.4.2

Multiply $5$ by $- 20$ .

$y = \frac{- 19 \pm \sqrt{361 + 180 - 100 x^{2} - 20 (10 x)}}{2 \cdot 5}$

Step 9.4.1.4.3

Multiply $10$ by $- 20$ .

$y = \frac{- 19 \pm \sqrt{361 + 180 - 100 x^{2} - 200 x}}{2 \cdot 5}$

Step 9.4.1.5

Add $361$ and $180$ .

$y = \frac{- 19 \pm \sqrt{541 - 100 x^{2} - 200 x}}{2 \cdot 5}$

Step 9.4.1.6

Reorder terms.

$y = \frac{- 19 \pm \sqrt{- 100 x^{2} - 200 x + 541}}{2 \cdot 5}$

Step 9.4.2

Multiply $2$ by $5$ .

$y = \frac{- 19 \pm \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 9.5

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

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Step 9.5.1

Simplify the numerator.

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Step 9.5.1.1

Raise $19$ to the power of $2$ .

$y = \frac{- 19 \pm \sqrt{361 - 4 \cdot 5 \cdot (- 9 + 5 x^{2} + 10 x)}}{2 \cdot 5}$

Step 9.5.1.2

Multiply $- 4$ by $5$ .

$y = \frac{- 19 \pm \sqrt{361 - 20 \cdot (- 9 + 5 x^{2} + 10 x)}}{2 \cdot 5}$

Step 9.5.1.3

Apply the distributive property.

$y = \frac{- 19 \pm \sqrt{361 - 20 \cdot - 9 - 20 (5 x^{2}) - 20 (10 x)}}{2 \cdot 5}$

Step 9.5.1.4

Simplify.

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Step 9.5.1.4.1

Multiply $- 20$ by $- 9$ .

$y = \frac{- 19 \pm \sqrt{361 + 180 - 20 (5 x^{2}) - 20 (10 x)}}{2 \cdot 5}$

Step 9.5.1.4.2

Multiply $5$ by $- 20$ .

$y = \frac{- 19 \pm \sqrt{361 + 180 - 100 x^{2} - 20 (10 x)}}{2 \cdot 5}$

Step 9.5.1.4.3

Multiply $10$ by $- 20$ .

$y = \frac{- 19 \pm \sqrt{361 + 180 - 100 x^{2} - 200 x}}{2 \cdot 5}$

Step 9.5.1.5

Add $361$ and $180$ .

$y = \frac{- 19 \pm \sqrt{541 - 100 x^{2} - 200 x}}{2 \cdot 5}$

Step 9.5.1.6

Reorder terms.

$y = \frac{- 19 \pm \sqrt{- 100 x^{2} - 200 x + 541}}{2 \cdot 5}$

Step 9.5.2

Multiply $2$ by $5$ .

$y = \frac{- 19 \pm \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 9.5.3

Change the $\pm$ to $+$ .

$y = \frac{- 19 + \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 9.5.4

Rewrite $- 19$ as $- 1 (19)$ .

$y = \frac{- 1 \cdot 19 + \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 9.5.5

Factor $- 1$ out of $\sqrt{- 100 x^{2} - 200 x + 541}$ .

$y = \frac{- 1 \cdot 19 - 1 (- \sqrt{- 100 x^{2} - 200 x + 541})}{10}$

Step 9.5.6

Factor $- 1$ out of $- 1 (19) - 1 (- \sqrt{- 100 x^{2} - 200 x + 541})$ .

$y = \frac{- 1 (19 - \sqrt{- 100 x^{2} - 200 x + 541})}{10}$

Step 9.5.7

Move the negative in front of the fraction.

$y = - \frac{19 - \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 9.6

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

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Step 9.6.1

Simplify the numerator.

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Step 9.6.1.1

Raise $19$ to the power of $2$ .

$y = \frac{- 19 \pm \sqrt{361 - 4 \cdot 5 \cdot (- 9 + 5 x^{2} + 10 x)}}{2 \cdot 5}$

Step 9.6.1.2

Multiply $- 4$ by $5$ .

$y = \frac{- 19 \pm \sqrt{361 - 20 \cdot (- 9 + 5 x^{2} + 10 x)}}{2 \cdot 5}$

Step 9.6.1.3

Apply the distributive property.

$y = \frac{- 19 \pm \sqrt{361 - 20 \cdot - 9 - 20 (5 x^{2}) - 20 (10 x)}}{2 \cdot 5}$

Step 9.6.1.4

Simplify.

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Step 9.6.1.4.1

Multiply $- 20$ by $- 9$ .

$y = \frac{- 19 \pm \sqrt{361 + 180 - 20 (5 x^{2}) - 20 (10 x)}}{2 \cdot 5}$

Step 9.6.1.4.2

Multiply $5$ by $- 20$ .

$y = \frac{- 19 \pm \sqrt{361 + 180 - 100 x^{2} - 20 (10 x)}}{2 \cdot 5}$

Step 9.6.1.4.3

Multiply $10$ by $- 20$ .

$y = \frac{- 19 \pm \sqrt{361 + 180 - 100 x^{2} - 200 x}}{2 \cdot 5}$

Step 9.6.1.5

Add $361$ and $180$ .

$y = \frac{- 19 \pm \sqrt{541 - 100 x^{2} - 200 x}}{2 \cdot 5}$

Step 9.6.1.6

Reorder terms.

$y = \frac{- 19 \pm \sqrt{- 100 x^{2} - 200 x + 541}}{2 \cdot 5}$

Step 9.6.2

Multiply $2$ by $5$ .

$y = \frac{- 19 \pm \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 9.6.3

Change the $\pm$ to $-$ .

$y = \frac{- 19 - \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 9.6.4

Factor $- 1$ out of $- 19 - \sqrt{- 100 x^{2} - 200 x + 541}$ .

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Step 9.6.4.1

Rewrite $- 19$ as $- 1 (19)$ .

$y = \frac{- 1 \cdot 19 - \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 9.6.4.2

Factor $- 1$ out of $- \sqrt{- 100 x^{2} - 200 x + 541}$ .

$y = \frac{- 1 \cdot 19 - (\sqrt{- 100 x^{2} - 200 x + 541})}{10}$

Step 9.6.4.3

Factor $- 1$ out of $- 1 (19) - (\sqrt{- 100 x^{2} - 200 x + 541})$ .

$y = \frac{- 1 (19 + \sqrt{- 100 x^{2} - 200 x + 541})}{10}$

Step 9.6.4.4

Rewrite $- 1 (19 + \sqrt{- 100 x^{2} - 200 x + 541})$ as $- (19 + \sqrt{- 100 x^{2} - 200 x + 541})$ .

$y = \frac{- (19 + \sqrt{- 100 x^{2} - 200 x + 541})}{10}$

Step 9.6.5

Move the negative in front of the fraction.

$y = - \frac{19 + \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 9.7

The final answer is the combination of both solutions.

$y = - \frac{19 - \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

$y = - \frac{19 + \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

$y = - \frac{19 - \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

$y = - \frac{19 + \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 10

Assume that $y = x^{2}$ is $f (x) = x^{2}$ and $5 y^{2} + 19 y = 9 - 5 x^{2} - 10 x$ is $g (x) = - \frac{19 - \sqrt{- 100 x^{2} - 200 x + 541}}{10}, - \frac{19 + \sqrt{- 100 x^{2} - 200 x + 541}}{10}$ .

$f (x) = x^{2}$

$g (x) = - \frac{19 - \sqrt{- 100 x^{2} - 200 x + 541}}{10}, - \frac{19 + \sqrt{- 100 x^{2} - 200 x + 541}}{10}$

Step 11

The given functions are from different types. Transforming a function does not change its type, so it is impossible to transform $f (x) = x^{2}$ to $g (x) = - \frac{19 - \sqrt{- 100 x^{2} - 200 x + 541}}{10}$ .

Impossible geometric transformation

Step 12