Basic Math Examples

$2 y^{3} + 5 y^{2} = 8 y + 20$

Step 1

Subtract $8 y$ from both sides of the equation.

$2 y^{3} + 5 y^{2} - 8 y = 20$

Step 2

Subtract $20$ from both sides of the equation.

$2 y^{3} + 5 y^{2} - 8 y - 20 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor out the greatest common factor from each group.

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Step 3.1.1

Group the first two terms and the last two terms.

$(2 y^{3} + 5 y^{2}) - 8 y - 20 = 0$

Step 3.1.2

Factor out the greatest common factor (GCF) from each group.

$y^{2} (2 y + 5) - 4 (2 y + 5) = 0$

Step 3.2

Factor the polynomial by factoring out the greatest common factor, $2 y + 5$ .

$(2 y + 5) (y^{2} - 4) = 0$

Step 3.3

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

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

Step 3.4

Factor.

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Step 3.4.1

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

$(2 y + 5) ((y + 2) (y - 2)) = 0$

Step 3.4.2

Remove unnecessary parentheses.

$(2 y + 5) (y + 2) (y - 2) = 0$

Step 4

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

$2 y + 5 = 0$

$y + 2 = 0$

$y - 2 = 0$

Step 5

Set $2 y + 5$ equal to $0$ and solve for $y$ .

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Step 5.1

Set $2 y + 5$ equal to $0$ .

$2 y + 5 = 0$

Step 5.2

Solve $2 y + 5 = 0$ for $y$ .

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Step 5.2.1

Subtract $5$ from both sides of the equation.

$2 y = - 5$

Step 5.2.2

Divide each term in $2 y = - 5$ by $2$ and simplify.

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Step 5.2.2.1

Divide each term in $2 y = - 5$ by $2$ .

$\frac{2 y}{2} = \frac{- 5}{2}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{- 5}{2}$

Step 5.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 5}{2}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Move the negative in front of the fraction.

$y = - \frac{5}{2}$

Step 6

Set $y + 2$ equal to $0$ and solve for $y$ .

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Step 6.1

Set $y + 2$ equal to $0$ .

$y + 2 = 0$

Step 6.2

Subtract $2$ from both sides of the equation.

$y = - 2$

Step 7

Set $y - 2$ equal to $0$ and solve for $y$ .

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Step 7.1

Set $y - 2$ equal to $0$ .

$y - 2 = 0$

Step 7.2

Add $2$ to both sides of the equation.

$y = 2$

Step 8

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

$y = - \frac{5}{2}, - 2, 2$

Step 9

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{5}{2}, - 2, 2$

Decimal Form:

$y = - 2.5, - 2, 2$

Mixed Number Form:

$y = - 2 \frac{1}{2}, - 2, 2$