Pre-Algebra Examples

$5 x^{2} y + 8 x y^{2} = - 2$

Step 1

Add $2$ to both sides of the equation.

$5 x^{2} y + 8 x y^{2} + 2 = 0$

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 $a = 8 x$ , $b = 5 x^{2}$ , and $c = 2$ into the quadratic formula and solve for $y$ .

$\frac{- 5 x^{2} \pm \sqrt{{(5 x^{2})}^{2} - 4 \cdot (8 x \cdot 2)}}{2 (8 x)}$

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 $5 x^{2}$ .

$y = \frac{- 5 x^{2} \pm \sqrt{5^{2} {(x^{2})}^{2} - 4 \cdot (8 x) \cdot 2}}{2 \cdot (8 x)}$

Step 4.1.2

Raise $5$ to the power of $2$ .

$y = \frac{- 5 x^{2} \pm \sqrt{25 {(x^{2})}^{2} - 4 \cdot (8 x) \cdot 2}}{2 \cdot (8 x)}$

Step 4.1.3

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

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Step 4.1.3.1

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

$y = \frac{- 5 x^{2} \pm \sqrt{25 x^{2 \cdot 2} - 4 \cdot (8 x) \cdot 2}}{2 \cdot (8 x)}$

Step 4.1.3.2

Multiply $2$ by $2$ .

$y = \frac{- 5 x^{2} \pm \sqrt{25 x^{4} - 4 \cdot (8 x) \cdot 2}}{2 \cdot (8 x)}$

Step 4.1.4

Multiply $- 4$ by $8$ .

$y = \frac{- 5 x^{2} \pm \sqrt{25 x^{4} - 32 x \cdot 2}}{2 \cdot (8 x)}$

Step 4.1.5

Multiply $2$ by $- 32$ .

$y = \frac{- 5 x^{2} \pm \sqrt{25 x^{4} - 64 x}}{2 \cdot (8 x)}$

Step 4.1.6

Factor $x$ out of $25 x^{4} - 64 x$ .

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Step 4.1.6.1

Factor $x$ out of $25 x^{4}$ .

$y = \frac{- 5 x^{2} \pm \sqrt{x (25 x^{3}) - 64 x}}{2 \cdot (8 x)}$

Step 4.1.6.2

Factor $x$ out of $- 64 x$ .

$y = \frac{- 5 x^{2} \pm \sqrt{x (25 x^{3}) + x \cdot - 64}}{2 \cdot (8 x)}$

Step 4.1.6.3

Factor $x$ out of $x (25 x^{3}) + x \cdot - 64$ .

$y = \frac{- 5 x^{2} \pm \sqrt{x (25 x^{3} - 64)}}{2 \cdot (8 x)}$

Step 4.2

Multiply $2$ by $8$ .

$y = \frac{- 5 x^{2} \pm \sqrt{x (25 x^{3} - 64)}}{16 x}$

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

Apply the product rule to $5 x^{2}$ .

$y = \frac{- 5 x^{2} \pm \sqrt{5^{2} {(x^{2})}^{2} - 4 \cdot (8 x) \cdot 2}}{2 \cdot (8 x)}$

Step 5.1.2

Raise $5$ to the power of $2$ .

$y = \frac{- 5 x^{2} \pm \sqrt{25 {(x^{2})}^{2} - 4 \cdot (8 x) \cdot 2}}{2 \cdot (8 x)}$

Step 5.1.3

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

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Step 5.1.3.1

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

$y = \frac{- 5 x^{2} \pm \sqrt{25 x^{2 \cdot 2} - 4 \cdot (8 x) \cdot 2}}{2 \cdot (8 x)}$

Step 5.1.3.2

Multiply $2$ by $2$ .

$y = \frac{- 5 x^{2} \pm \sqrt{25 x^{4} - 4 \cdot (8 x) \cdot 2}}{2 \cdot (8 x)}$

Step 5.1.4

Multiply $- 4$ by $8$ .

$y = \frac{- 5 x^{2} \pm \sqrt{25 x^{4} - 32 x \cdot 2}}{2 \cdot (8 x)}$

Step 5.1.5

Multiply $2$ by $- 32$ .

$y = \frac{- 5 x^{2} \pm \sqrt{25 x^{4} - 64 x}}{2 \cdot (8 x)}$

Step 5.1.6

Factor $x$ out of $25 x^{4} - 64 x$ .

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Step 5.1.6.1

Factor $x$ out of $25 x^{4}$ .

$y = \frac{- 5 x^{2} \pm \sqrt{x (25 x^{3}) - 64 x}}{2 \cdot (8 x)}$

Step 5.1.6.2

Factor $x$ out of $- 64 x$ .

$y = \frac{- 5 x^{2} \pm \sqrt{x (25 x^{3}) + x \cdot - 64}}{2 \cdot (8 x)}$

Step 5.1.6.3

Factor $x$ out of $x (25 x^{3}) + x \cdot - 64$ .

$y = \frac{- 5 x^{2} \pm \sqrt{x (25 x^{3} - 64)}}{2 \cdot (8 x)}$

Step 5.2

Multiply $2$ by $8$ .

$y = \frac{- 5 x^{2} \pm \sqrt{x (25 x^{3} - 64)}}{16 x}$

Step 5.3

Change the $\pm$ to $+$ .

$y = \frac{- 5 x^{2} + \sqrt{x (25 x^{3} - 64)}}{16 x}$

Step 5.4

Factor $- 1$ out of $- 5 x^{2}$ .

$y = \frac{- (5 x^{2}) + \sqrt{x (25 x^{3} - 64)}}{16 x}$

Step 5.5

Factor $- 1$ out of $\sqrt{x (25 x^{3} - 64)}$ .

$y = \frac{- (5 x^{2}) - 1 (- \sqrt{x (25 x^{3} - 64)})}{16 x}$

Step 5.6

Factor $- 1$ out of $- (5 x^{2}) - 1 (- \sqrt{x (25 x^{3} - 64)})$ .

$y = \frac{- (5 x^{2} - \sqrt{x (25 x^{3} - 64)})}{16 x}$

Step 5.7

Rewrite $- (5 x^{2} - \sqrt{x (25 x^{3} - 64)})$ as $- 1 (5 x^{2} - \sqrt{x (25 x^{3} - 64)})$ .

$y = \frac{- 1 (5 x^{2} - \sqrt{x (25 x^{3} - 64)})}{16 x}$

Step 5.8

Move the negative in front of the fraction.

$y = - \frac{5 x^{2} - \sqrt{x (25 x^{3} - 64)}}{16 x}$

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

Apply the product rule to $5 x^{2}$ .

$y = \frac{- 5 x^{2} \pm \sqrt{5^{2} {(x^{2})}^{2} - 4 \cdot (8 x) \cdot 2}}{2 \cdot (8 x)}$

Step 6.1.2

Raise $5$ to the power of $2$ .

$y = \frac{- 5 x^{2} \pm \sqrt{25 {(x^{2})}^{2} - 4 \cdot (8 x) \cdot 2}}{2 \cdot (8 x)}$

Step 6.1.3

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

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Step 6.1.3.1

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

$y = \frac{- 5 x^{2} \pm \sqrt{25 x^{2 \cdot 2} - 4 \cdot (8 x) \cdot 2}}{2 \cdot (8 x)}$

Step 6.1.3.2

Multiply $2$ by $2$ .

$y = \frac{- 5 x^{2} \pm \sqrt{25 x^{4} - 4 \cdot (8 x) \cdot 2}}{2 \cdot (8 x)}$

Step 6.1.4

Multiply $- 4$ by $8$ .

$y = \frac{- 5 x^{2} \pm \sqrt{25 x^{4} - 32 x \cdot 2}}{2 \cdot (8 x)}$

Step 6.1.5

Multiply $2$ by $- 32$ .

$y = \frac{- 5 x^{2} \pm \sqrt{25 x^{4} - 64 x}}{2 \cdot (8 x)}$

Step 6.1.6

Factor $x$ out of $25 x^{4} - 64 x$ .

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Step 6.1.6.1

Factor $x$ out of $25 x^{4}$ .

$y = \frac{- 5 x^{2} \pm \sqrt{x (25 x^{3}) - 64 x}}{2 \cdot (8 x)}$

Step 6.1.6.2

Factor $x$ out of $- 64 x$ .

$y = \frac{- 5 x^{2} \pm \sqrt{x (25 x^{3}) + x \cdot - 64}}{2 \cdot (8 x)}$

Step 6.1.6.3

Factor $x$ out of $x (25 x^{3}) + x \cdot - 64$ .

$y = \frac{- 5 x^{2} \pm \sqrt{x (25 x^{3} - 64)}}{2 \cdot (8 x)}$

Step 6.2

Multiply $2$ by $8$ .

$y = \frac{- 5 x^{2} \pm \sqrt{x (25 x^{3} - 64)}}{16 x}$

Step 6.3

Change the $\pm$ to $-$ .

$y = \frac{- 5 x^{2} - \sqrt{x (25 x^{3} - 64)}}{16 x}$

Step 6.4

Factor $- 1$ out of $- 5 x^{2}$ .

$y = \frac{- (5 x^{2}) - \sqrt{x (25 x^{3} - 64)}}{16 x}$

Step 6.5

Factor $- 1$ out of $- \sqrt{x (25 x^{3} - 64)}$ .

$y = \frac{- (5 x^{2}) - (\sqrt{x (25 x^{3} - 64)})}{16 x}$

Step 6.6

Factor $- 1$ out of $- (5 x^{2}) - (\sqrt{x (25 x^{3} - 64)})$ .

$y = \frac{- (5 x^{2} + \sqrt{x (25 x^{3} - 64)})}{16 x}$

Step 6.7

Rewrite $- (5 x^{2} + \sqrt{x (25 x^{3} - 64)})$ as $- 1 (5 x^{2} + \sqrt{x (25 x^{3} - 64)})$ .

$y = \frac{- 1 (5 x^{2} + \sqrt{x (25 x^{3} - 64)})}{16 x}$

Step 6.8

Move the negative in front of the fraction.

$y = - \frac{5 x^{2} + \sqrt{x (25 x^{3} - 64)}}{16 x}$

Step 7

The final answer is the combination of both solutions.

$y = - \frac{5 x^{2} - \sqrt{x (25 x^{3} - 64)}}{16 x}$

$y = - \frac{5 x^{2} + \sqrt{x (25 x^{3} - 64)}}{16 x}$

Step 8

The given equation $y = - \frac{5 x^{2} - \sqrt{x (25 x^{3} - 64)}}{16 x}, - \frac{5 x^{2} + \sqrt{x (25 x^{3} - 64)}}{16 x}$ can not be written as $y = k x^{2}$ , so $y$ doesn't vary directly with $x^{2}$ .

$y$ doesn't vary directly with $x$