Calculus Examples

$5 X^{2} y + 7 X y^{2} = 9$

Step 1

Subtract $9$ from both sides of the equation.

$5 X^{2} y + 7 X y^{2} - 9 = 0$

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 $a = 5 y$ , $b = 7 y^{2}$ , and $c = - 9$ into the quadratic formula and solve for $X$ .

$\frac{- 7 y^{2} \pm \sqrt{{(7 y^{2})}^{2} - 4 \cdot (5 y \cdot - 9)}}{2 (5 y)}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the product rule to $7 y^{2}$ .

$X = \frac{- 7 y^{2} \pm \sqrt{7^{2} {(y^{2})}^{2} - 4 \cdot (5 y) \cdot - 9}}{2 \cdot (5 y)}$

Step 4.1.2

Raise $7$ to the power of $2$ .

$X = \frac{- 7 y^{2} \pm \sqrt{49 {(y^{2})}^{2} - 4 \cdot (5 y) \cdot - 9}}{2 \cdot (5 y)}$

Step 4.1.3

Multiply the exponents in ${(y^{2})}^{2}$ .

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Step 4.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$X = \frac{- 7 y^{2} \pm \sqrt{49 y^{2 \cdot 2} - 4 \cdot (5 y) \cdot - 9}}{2 \cdot (5 y)}$

Step 4.1.3.2

Multiply $2$ by $2$ .

$X = \frac{- 7 y^{2} \pm \sqrt{49 y^{4} - 4 \cdot (5 y) \cdot - 9}}{2 \cdot (5 y)}$

Step 4.1.4

Multiply $- 4$ by $5$ .

$X = \frac{- 7 y^{2} \pm \sqrt{49 y^{4} - 20 y \cdot - 9}}{2 \cdot (5 y)}$

Step 4.1.5

Multiply $- 9$ by $- 20$ .

$X = \frac{- 7 y^{2} \pm \sqrt{49 y^{4} + 180 y}}{2 \cdot (5 y)}$

Step 4.1.6

Factor $y$ out of $49 y^{4} + 180 y$ .

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Step 4.1.6.1

Factor $y$ out of $49 y^{4}$ .

$X = \frac{- 7 y^{2} \pm \sqrt{y (49 y^{3}) + 180 y}}{2 \cdot (5 y)}$

Step 4.1.6.2

Factor $y$ out of $180 y$ .

$X = \frac{- 7 y^{2} \pm \sqrt{y (49 y^{3}) + y \cdot 180}}{2 \cdot (5 y)}$

Step 4.1.6.3

Factor $y$ out of $y (49 y^{3}) + y \cdot 180$ .

$X = \frac{- 7 y^{2} \pm \sqrt{y (49 y^{3} + 180)}}{2 \cdot (5 y)}$

Step 4.2

Multiply $2$ by $5$ .

$X = \frac{- 7 y^{2} \pm \sqrt{y (49 y^{3} + 180)}}{10 y}$

Step 5

The final answer is the combination of both solutions.

$X = - \frac{7 y^{2} - \sqrt{y (49 y^{3} + 180)}}{10 y}$

$X = - \frac{7 y^{2} + \sqrt{y (49 y^{3} + 180)}}{10 y}$