Calculus Examples

$3 y^{3} + (1) y - y = 375 {(1)}^{4}$

Step 1

Simplify $3 y^{3} + 1 y - y$ .

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Step 1.1

Multiply $y$ by $1$ .

$3 y^{3} + y - y = 375 \cdot 1^{4}$

Step 1.2

Combine the opposite terms in $3 y^{3} + y - y$ .

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Step 1.2.1

Subtract $y$ from $y$ .

$3 y^{3} + 0 = 375 \cdot 1^{4}$

Step 1.2.2

Add $3 y^{3}$ and $0$ .

$3 y^{3} = 375 \cdot 1^{4}$

Step 2

Simplify $375 \cdot 1^{4}$ .

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Step 2.1

One to any power is one.

$3 y^{3} = 375 \cdot 1$

Step 2.2

Multiply $375$ by $1$ .

$3 y^{3} = 375$

Step 3

Subtract $375$ from both sides of the equation.

$3 y^{3} - 375 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $3$ out of $3 y^{3} - 375$ .

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Step 4.1.1

Factor $3$ out of $3 y^{3}$ .

$3 (y^{3}) - 375 = 0$

Step 4.1.2

Factor $3$ out of $- 375$ .

$3 y^{3} + 3 \cdot - 125 = 0$

Step 4.1.3

Factor $3$ out of $3 y^{3} + 3 \cdot - 125$ .

$3 (y^{3} - 125) = 0$

Step 4.2

Rewrite $125$ as $5^{3}$ .

$3 (y^{3} - 5^{3}) = 0$

Step 4.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = y$ and $b = 5$ .

$3 ((y - 5) (y^{2} + y \cdot 5 + 5^{2})) = 0$

Step 4.4

Factor.

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Step 4.4.1

Simplify.

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Step 4.4.1.1

Move $5$ to the left of $y$ .

$3 ((y - 5) (y^{2} + 5 y + 5^{2})) = 0$

Step 4.4.1.2

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

$3 ((y - 5) (y^{2} + 5 y + 25)) = 0$

Step 4.4.2

Remove unnecessary parentheses.

$3 (y - 5) (y^{2} + 5 y + 25) = 0$

Step 5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 5 = 0$

$y^{2} + 5 y + 25 = 0$

Step 6

Set $y - 5$ equal to $0$ and solve for $y$ .

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Step 6.1

Set $y - 5$ equal to $0$ .

$y - 5 = 0$

Step 6.2

Add $5$ to both sides of the equation.

$y = 5$

Step 7

Set $y^{2} + 5 y + 25$ equal to $0$ and solve for $y$ .

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Step 7.1

Set $y^{2} + 5 y + 25$ equal to $0$ .

$y^{2} + 5 y + 25 = 0$

Step 7.2

Solve $y^{2} + 5 y + 25 = 0$ for $y$ .

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Step 7.2.1

Use the quadratic formula to find the solutions.

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

Step 7.2.2

Substitute the values $a = 1$ , $b = 5$ , and $c = 25$ into the quadratic formula and solve for $y$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (1 \cdot 25)}}{2 \cdot 1}$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Simplify the numerator.

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Step 7.2.3.1.1

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

$y = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot 25}}{2 \cdot 1}$

Step 7.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 25$ .

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Step 7.2.3.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{- 5 \pm \sqrt{25 - 4 \cdot 25}}{2 \cdot 1}$

Step 7.2.3.1.2.2

Multiply $- 4$ by $25$ .

$y = \frac{- 5 \pm \sqrt{25 - 100}}{2 \cdot 1}$

Step 7.2.3.1.3

Subtract $100$ from $25$ .

$y = \frac{- 5 \pm \sqrt{- 75}}{2 \cdot 1}$

Step 7.2.3.1.4

Rewrite $- 75$ as $- 1 (75)$ .

$y = \frac{- 5 \pm \sqrt{- 1 \cdot 75}}{2 \cdot 1}$

Step 7.2.3.1.5

Rewrite $\sqrt{- 1 (75)}$ as $\sqrt{- 1} \cdot \sqrt{75}$ .

$y = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{75}}{2 \cdot 1}$

Step 7.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{- 5 \pm i \cdot \sqrt{75}}{2 \cdot 1}$

Step 7.2.3.1.7

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 7.2.3.1.7.1

Factor $25$ out of $75$ .

$y = \frac{- 5 \pm i \cdot \sqrt{25 (3)}}{2 \cdot 1}$

Step 7.2.3.1.7.2

Rewrite $25$ as $5^{2}$ .

$y = \frac{- 5 \pm i \cdot \sqrt{5^{2} \cdot 3}}{2 \cdot 1}$

Step 7.2.3.1.8

Pull terms out from under the radical.

$y = \frac{- 5 \pm i \cdot (5 \sqrt{3})}{2 \cdot 1}$

Step 7.2.3.1.9

Move $5$ to the left of $i$ .

$y = \frac{- 5 \pm 5 i \sqrt{3}}{2 \cdot 1}$

Step 7.2.3.2

Multiply $2$ by $1$ .

$y = \frac{- 5 \pm 5 i \sqrt{3}}{2}$

Step 7.2.4

The final answer is the combination of both solutions.

$y = - \frac{5 - 5 i \sqrt{3}}{2}, - \frac{5 + 5 i \sqrt{3}}{2}$

Step 8

The final solution is all the values that make $3 (y - 5) (y^{2} + 5 y + 25) = 0$ true.

$y = 5, - \frac{5 - 5 i \sqrt{3}}{2}, - \frac{5 + 5 i \sqrt{3}}{2}$