Calculus Examples

$x^{3} - 2 x^{2} y + 3 x y^{2} = 38$

Step 1

Subtract $38$ from both sides of the equation.

$x^{3} - 2 x^{2} y + 3 x y^{2} - 38 = 0$

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 $a = 3 x$ , $b = - 2 x^{2}$ , and $c = x^{3} - 38$ into the quadratic formula and solve for $y$ .

$\frac{2 x^{2} \pm \sqrt{{(- 2 x^{2})}^{2} - 4 \cdot (3 x \cdot (x^{3} - 38))}}{2 (3 x)}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

Add parentheses.

$y = \frac{2 x^{2} \pm \sqrt{{(- 2 x^{2})}^{2} - 4 \cdot (3 (x \cdot (x^{3} - 38)))}}{2 \cdot (3 x)}$

Step 4.1.2

Let $u = 3 (x \cdot (x^{3} - 38))$ . Substitute $u$ for all occurrences of $3 (x \cdot (x^{3} - 38))$ .

Tap for more steps...

Step 4.1.2.1

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

$y = \frac{2 x^{2} \pm \sqrt{{(- 2)}^{2} {(x^{2})}^{2} - 4 \cdot u}}{2 \cdot (3 x)}$

Step 4.1.2.2

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

$y = \frac{2 x^{2} \pm \sqrt{4 {(x^{2})}^{2} - 4 \cdot u}}{2 \cdot (3 x)}$

Step 4.1.2.3

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

Tap for more steps...

Step 4.1.2.3.1

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

$y = \frac{2 x^{2} \pm \sqrt{4 x^{2 \cdot 2} - 4 \cdot u}}{2 \cdot (3 x)}$

Step 4.1.2.3.2

Multiply $2$ by $2$ .

$y = \frac{2 x^{2} \pm \sqrt{4 x^{4} - 4 \cdot u}}{2 \cdot (3 x)}$

$y = \frac{2 x^{2} \pm \sqrt{4 x^{4} - 4 u}}{2 \cdot (3 x)}$

Step 4.1.3

Factor $4$ out of $4 x^{4} - 4 u$ .

Tap for more steps...

Step 4.1.3.1

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

$y = \frac{2 x^{2} \pm \sqrt{4 (x^{4}) - 4 u}}{2 \cdot (3 x)}$

Step 4.1.3.2

Factor $4$ out of $- 4 u$ .

$y = \frac{2 x^{2} \pm \sqrt{4 (x^{4}) + 4 (- u)}}{2 \cdot (3 x)}$

Step 4.1.3.3

Factor $4$ out of $4 (x^{4}) + 4 (- u)$ .

$y = \frac{2 x^{2} \pm \sqrt{4 (x^{4} - u)}}{2 \cdot (3 x)}$

Step 4.1.4

Replace all occurrences of $u$ with $3 (x \cdot (x^{3} - 38))$ .

$y = \frac{2 x^{2} \pm \sqrt{4 (x^{4} - (3 (x \cdot (x^{3} - 38))))}}{2 \cdot (3 x)}$

Step 4.1.5

Simplify.

Tap for more steps...

Step 4.1.5.1

Simplify each term.

Tap for more steps...

Step 4.1.5.1.1

Apply the distributive property.

$y = \frac{2 x^{2} \pm \sqrt{4 (x^{4} - (3 (x \cdot x^{3} + x \cdot - 38)))}}{2 \cdot (3 x)}$

Step 4.1.5.1.2

Multiply $x$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 4.1.5.1.2.1

Multiply $x$ by $x^{3}$ .

Tap for more steps...

Step 4.1.5.1.2.1.1

Raise $x$ to the power of $1$ .

$y = \frac{2 x^{2} \pm \sqrt{4 (x^{4} - (3 (x \cdot x^{3} + x \cdot - 38)))}}{2 \cdot (3 x)}$

Step 4.1.5.1.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{2 x^{2} \pm \sqrt{4 (x^{4} - (3 (x^{1 + 3} + x \cdot - 38)))}}{2 \cdot (3 x)}$

Step 4.1.5.1.2.2

Add $1$ and $3$ .

$y = \frac{2 x^{2} \pm \sqrt{4 (x^{4} - (3 (x^{4} + x \cdot - 38)))}}{2 \cdot (3 x)}$

Step 4.1.5.1.3

Move $- 38$ to the left of $x$ .

$y = \frac{2 x^{2} \pm \sqrt{4 (x^{4} - (3 (x^{4} - 38 \cdot x)))}}{2 \cdot (3 x)}$

Step 4.1.5.1.4

Apply the distributive property.

$y = \frac{2 x^{2} \pm \sqrt{4 (x^{4} - (3 x^{4} + 3 (- 38 x)))}}{2 \cdot (3 x)}$

Step 4.1.5.1.5

Multiply $- 38$ by $3$ .

$y = \frac{2 x^{2} \pm \sqrt{4 (x^{4} - (3 x^{4} - 114 x))}}{2 \cdot (3 x)}$

Step 4.1.5.1.6

Apply the distributive property.

$y = \frac{2 x^{2} \pm \sqrt{4 (x^{4} - (3 x^{4}) - (- 114 x))}}{2 \cdot (3 x)}$

Step 4.1.5.1.7

Multiply $3$ by $- 1$ .

$y = \frac{2 x^{2} \pm \sqrt{4 (x^{4} - 3 x^{4} - (- 114 x))}}{2 \cdot (3 x)}$

Step 4.1.5.1.8

Multiply $- 114$ by $- 1$ .

$y = \frac{2 x^{2} \pm \sqrt{4 (x^{4} - 3 x^{4} + 114 x)}}{2 \cdot (3 x)}$

Step 4.1.5.2

Subtract $3 x^{4}$ from $x^{4}$ .

$y = \frac{2 x^{2} \pm \sqrt{4 (- 2 x^{4} + 114 x)}}{2 \cdot (3 x)}$

Step 4.1.6

Factor $2 x$ out of $- 2 x^{4} + 114 x$ .

Tap for more steps...

Step 4.1.6.1

Factor $2 x$ out of $- 2 x^{4}$ .

$y = \frac{2 x^{2} \pm \sqrt{4 (2 x (- x^{3}) + 114 x)}}{2 \cdot (3 x)}$

Step 4.1.6.2

Factor $2 x$ out of $114 x$ .

$y = \frac{2 x^{2} \pm \sqrt{4 (2 x (- x^{3}) + 2 x (57))}}{2 \cdot (3 x)}$

Step 4.1.6.3

Factor $2 x$ out of $2 x (- x^{3}) + 2 x (57)$ .

$y = \frac{2 x^{2} \pm \sqrt{4 (2 x (- x^{3} + 57))}}{2 \cdot (3 x)}$

$y = \frac{2 x^{2} \pm \sqrt{4 \cdot (2 x (- x^{3} + 57))}}{2 \cdot (3 x)}$

Step 4.1.7

Multiply $4$ by $2$ .

$y = \frac{2 x^{2} \pm \sqrt{8 x (- x^{3} + 57)}}{2 \cdot (3 x)}$

Step 4.1.8

Rewrite $8 x (- x^{3} + 57)$ as $2^{2} \cdot (2 x (- x^{3} + 57))$ .

Tap for more steps...

Step 4.1.8.1

Factor $4$ out of $8$ .

$y = \frac{2 x^{2} \pm \sqrt{4 (2) x (- x^{3} + 57)}}{2 \cdot (3 x)}$

Step 4.1.8.2

Rewrite $4$ as $2^{2}$ .

$y = \frac{2 x^{2} \pm \sqrt{2^{2} \cdot (2 x (- x^{3} + 57))}}{2 \cdot (3 x)}$

Step 4.1.8.3

Add parentheses.

$y = \frac{2 x^{2} \pm \sqrt{2^{2} \cdot (2 (x (- x^{3} + 57)))}}{2 \cdot (3 x)}$

Step 4.1.8.4

Add parentheses.

$y = \frac{2 x^{2} \pm \sqrt{2^{2} \cdot (2 x (- x^{3} + 57))}}{2 \cdot (3 x)}$

Step 4.1.9

Pull terms out from under the radical.

$y = \frac{2 x^{2} \pm 2 \sqrt{2 x (- x^{3} + 57)}}{2 \cdot (3 x)}$

Step 4.2

Multiply $2$ by $3$ .

$y = \frac{2 x^{2} \pm 2 \sqrt{2 x (- x^{3} + 57)}}{6 x}$

Step 4.3

Simplify $\frac{2 x^{2} \pm 2 \sqrt{2 x (- x^{3} + 57)}}{6 x}$ .

$y = \frac{x^{2} \pm \sqrt{2 x (- x^{3} + 57)}}{3 x}$

Step 5

The final answer is the combination of both solutions.

$y = \frac{x^{2} + \sqrt{2 x (- x^{3} + 57)}}{3 x}$

$y = \frac{x^{2} - \sqrt{2 x (- x^{3} + 57)}}{3 x}$