Precalculus Examples

$49 x^{2} + 686 + 36 y^{2} = 0$

Step 1

Solve for $y$ .

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Step 1.1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1.1

Subtract $49 x^{2}$ from both sides of the equation.

$686 + 36 y^{2} = - 49 x^{2}$

Step 1.1.2

Subtract $686$ from both sides of the equation.

$36 y^{2} = - 49 x^{2} - 686$

Step 1.2

Divide each term in $36 y^{2} = - 49 x^{2} - 686$ by $36$ and simplify.

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Step 1.2.1

Divide each term in $36 y^{2} = - 49 x^{2} - 686$ by $36$ .

$\frac{36 y^{2}}{36} = \frac{- 49 x^{2}}{36} + \frac{- 686}{36}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $36$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{36 y^{2}}{36} = \frac{- 49 x^{2}}{36} + \frac{- 686}{36}$

Step 1.2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{- 49 x^{2}}{36} + \frac{- 686}{36}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

$y^{2} = - \frac{49 x^{2}}{36} + \frac{- 686}{36}$

Step 1.2.3.1.2

Cancel the common factor of $- 686$ and $36$ .

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Step 1.2.3.1.2.1

Factor $2$ out of $- 686$ .

$y^{2} = - \frac{49 x^{2}}{36} + \frac{2 (- 343)}{36}$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $2$ out of $36$ .

$y^{2} = - \frac{49 x^{2}}{36} + \frac{2 \cdot - 343}{2 \cdot 18}$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$y^{2} = - \frac{49 x^{2}}{36} + \frac{2 \cdot - 343}{2 \cdot 18}$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$y^{2} = - \frac{49 x^{2}}{36} + \frac{- 343}{18}$

Step 1.2.3.1.3

Move the negative in front of the fraction.

$y^{2} = - \frac{49 x^{2}}{36} - \frac{343}{18}$

Step 1.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{- \frac{49 x^{2}}{36} - \frac{343}{18}}$

Step 1.4

Simplify $\pm \sqrt{- \frac{49 x^{2}}{36} - \frac{343}{18}}$ .

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Step 1.4.1

Factor $- 1$ out of $- \frac{49 x^{2}}{36} - \frac{343}{18}$ .

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Step 1.4.1.1

Factor $- 1$ out of $- \frac{49 x^{2}}{36}$ .

$y = \pm \sqrt{- (\frac{49 x^{2}}{36}) - \frac{343}{18}}$

Step 1.4.1.2

Factor $- 1$ out of $- \frac{343}{18}$ .

$y = \pm \sqrt{- (\frac{49 x^{2}}{36}) - (\frac{343}{18})}$

Step 1.4.1.3

Factor $- 1$ out of $- (\frac{49 x^{2}}{36}) - (\frac{343}{18})$ .

$y = \pm \sqrt{- (\frac{49 x^{2}}{36} + \frac{343}{18})}$

Step 1.4.2

To write $\frac{343}{18}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \pm \sqrt{- (\frac{49 x^{2}}{36} + \frac{343}{18} \cdot \frac{2}{2})}$

Step 1.4.3

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.3.1

Multiply $\frac{343}{18}$ by $\frac{2}{2}$ .

$y = \pm \sqrt{- (\frac{49 x^{2}}{36} + \frac{343 \cdot 2}{18 \cdot 2})}$

Step 1.4.3.2

Multiply $18$ by $2$ .

$y = \pm \sqrt{- (\frac{49 x^{2}}{36} + \frac{343 \cdot 2}{36})}$

Step 1.4.4

Combine the numerators over the common denominator.

$y = \pm \sqrt{- \frac{49 x^{2} + 343 \cdot 2}{36}}$

Step 1.4.5

Simplify the numerator.

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Step 1.4.5.1

Factor $49$ out of $49 x^{2} + 343 \cdot 2$ .

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Step 1.4.5.1.1

Factor $49$ out of $49 x^{2}$ .

$y = \pm \sqrt{- \frac{49 (x^{2}) + 343 \cdot 2}{36}}$

Step 1.4.5.1.2

Factor $49$ out of $343 \cdot 2$ .

$y = \pm \sqrt{- \frac{49 (x^{2}) + 49 (7 \cdot 2)}{36}}$

Step 1.4.5.1.3

Factor $49$ out of $49 (x^{2}) + 49 (7 \cdot 2)$ .

$y = \pm \sqrt{- \frac{49 (x^{2} + 7 \cdot 2)}{36}}$

Step 1.4.5.2

Multiply $7$ by $2$ .

$y = \pm \sqrt{- \frac{49 (x^{2} + 14)}{36}}$

Step 1.4.6

Rewrite $- \frac{49 (x^{2} + 14)}{36}$ as ${(\frac{7}{6})}^{2} \cdot (- (x^{2} + 14))$ .

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Step 1.4.6.1

Factor the perfect power $7^{2}$ out of $49 (x^{2} + 14)$ .

$y = \pm \sqrt{- \frac{7^{2} (x^{2} + 14)}{36}}$

Step 1.4.6.2

Factor the perfect power $6^{2}$ out of $36$ .

$y = \pm \sqrt{- \frac{7^{2} (x^{2} + 14)}{6^{2} \cdot 1}}$

Step 1.4.6.3

Rearrange the fraction $\frac{7^{2} (x^{2} + 14)}{6^{2} \cdot 1}$ .

$y = \pm \sqrt{- ({(\frac{7}{6})}^{2} (x^{2} + 14))}$

Step 1.4.6.4

Reorder $- 1$ and ${(\frac{7}{6})}^{2}$ .

$y = \pm \sqrt{{(\frac{7}{6})}^{2} \cdot - 1 (x^{2} + 14)}$

Step 1.4.6.5

Add parentheses.

$y = \pm \sqrt{{(\frac{7}{6})}^{2} \cdot (- (x^{2} + 14))}$

Step 1.4.7

Pull terms out from under the radical.

$y = \pm \frac{7}{6} \sqrt{- (x^{2} + 14)}$

Step 1.4.8

Combine $\frac{7}{6}$ and $\sqrt{- (x^{2} + 14)}$ .

$y = \pm \frac{7 \sqrt{- (x^{2} + 14)}}{6}$

Step 1.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{7 \sqrt{- (x^{2} + 14)}}{6}$

Step 1.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{7 \sqrt{- (x^{2} + 14)}}{6}$

Step 1.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{7 \sqrt{- (x^{2} + 14)}}{6}$

$y = - \frac{7 \sqrt{- (x^{2} + 14)}}{6}$

$y = \frac{7 \sqrt{- (x^{2} + 14)}}{6}$

$y = - \frac{7 \sqrt{- (x^{2} + 14)}}{6}$

$y = \frac{7 \sqrt{- (x^{2} + 14)}}{6}$

$y = - \frac{7 \sqrt{- (x^{2} + 14)}}{6}$

Step 2

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$y = a x^{2} + b x + c$

Step 3

Reorder terms.

$y = \frac{7}{6} \cdot \sqrt{- (x^{2} + 14)}$

$y = - (\frac{7}{6} \cdot \sqrt{- (x^{2} + 14)})$

Step 4

Remove parentheses.

$y = \frac{7}{6} \cdot \sqrt{- (x^{2} + 14)}$

$y = - \frac{7}{6} \cdot \sqrt{- (x^{2} + 14)}$

Step 5