Trigonometry Examples

$40 a^{2} + 20 a - 30 = (- 1) (15 a^{3} + 30)$

Step 1

Simplify $- 1 (15 a^{3} + 30)$ .

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Step 1.1

Rewrite.

$40 a^{2} + 20 a - 30 = 0 + 0 - 1 (15 a^{3} + 30)$

Step 1.2

Simplify by adding zeros.

$40 a^{2} + 20 a - 30 = - 1 (15 a^{3} + 30)$

Step 1.3

Apply the distributive property.

$40 a^{2} + 20 a - 30 = - 1 (15 a^{3}) - 1 \cdot 30$

Step 1.4

Multiply.

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Step 1.4.1

Multiply $15$ by $- 1$ .

$40 a^{2} + 20 a - 30 = - 15 a^{3} - 1 \cdot 30$

Step 1.4.2

Multiply $- 1$ by $30$ .

$40 a^{2} + 20 a - 30 = - 15 a^{3} - 30$

Step 2

Add $15 a^{3}$ to both sides of the equation.

$40 a^{2} + 20 a - 30 + 15 a^{3} = - 30$

Step 3

Add $30$ to both sides of the equation.

$40 a^{2} + 20 a - 30 + 15 a^{3} + 30 = 0$

Step 4

Combine the opposite terms in $40 a^{2} + 20 a - 30 + 15 a^{3} + 30$ .

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Step 4.1

Add $- 30$ and $30$ .

$40 a^{2} + 20 a + 15 a^{3} + 0 = 0$

Step 4.2

Add $40 a^{2} + 20 a + 15 a^{3}$ and $0$ .

$40 a^{2} + 20 a + 15 a^{3} = 0$

Step 5

Factor the left side of the equation.

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Step 5.1

Factor $5 a$ out of $40 a^{2} + 20 a + 15 a^{3}$ .

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Step 5.1.1

Factor $5 a$ out of $40 a^{2}$ .

$5 a (8 a) + 20 a + 15 a^{3} = 0$

Step 5.1.2

Factor $5 a$ out of $20 a$ .

$5 a (8 a) + 5 a (4) + 15 a^{3} = 0$

Step 5.1.3

Factor $5 a$ out of $15 a^{3}$ .

$5 a (8 a) + 5 a (4) + 5 a (3 a^{2}) = 0$

Step 5.1.4

Factor $5 a$ out of $5 a (8 a) + 5 a (4)$ .

$5 a (8 a + 4) + 5 a (3 a^{2}) = 0$

Step 5.1.5

Factor $5 a$ out of $5 a (8 a + 4) + 5 a (3 a^{2})$ .

$5 a (8 a + 4 + 3 a^{2}) = 0$

Step 5.2

Factor.

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Step 5.2.1

Factor by grouping.

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Step 5.2.1.1

Reorder terms.

$5 a (3 a^{2} + 8 a + 4) = 0$

Step 5.2.1.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 4 = 12$ and whose sum is $b = 8$ .

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Step 5.2.1.2.1

Factor $8$ out of $8 a$ .

$5 a (3 a^{2} + 8 (a) + 4) = 0$

Step 5.2.1.2.2

Rewrite $8$ as $2$ plus $6$

$5 a (3 a^{2} + (2 + 6) a + 4) = 0$

Step 5.2.1.2.3

Apply the distributive property.

$5 a (3 a^{2} + 2 a + 6 a + 4) = 0$

Step 5.2.1.3

Factor out the greatest common factor from each group.

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Step 5.2.1.3.1

Group the first two terms and the last two terms.

$5 a ((3 a^{2} + 2 a) + 6 a + 4) = 0$

Step 5.2.1.3.2

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

$5 a (a (3 a + 2) + 2 (3 a + 2)) = 0$

Step 5.2.1.4

Factor the polynomial by factoring out the greatest common factor, $3 a + 2$ .

$5 a ((3 a + 2) (a + 2)) = 0$

Step 5.2.2

Remove unnecessary parentheses.

$5 a (3 a + 2) (a + 2) = 0$

Step 6

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

$a = 0$

$3 a + 2 = 0$

$a + 2 = 0$

Step 7

Set $a$ equal to $0$ .

$a = 0$

Step 8

Set $3 a + 2$ equal to $0$ and solve for $a$ .

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Step 8.1

Set $3 a + 2$ equal to $0$ .

$3 a + 2 = 0$

Step 8.2

Solve $3 a + 2 = 0$ for $a$ .

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Step 8.2.1

Subtract $2$ from both sides of the equation.

$3 a = - 2$

Step 8.2.2

Divide each term in $3 a = - 2$ by $3$ and simplify.

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Step 8.2.2.1

Divide each term in $3 a = - 2$ by $3$ .

$\frac{3 a}{3} = \frac{- 2}{3}$

Step 8.2.2.2

Simplify the left side.

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Step 8.2.2.2.1

Cancel the common factor of $3$ .

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Step 8.2.2.2.1.1

Cancel the common factor.

$\frac{3 a}{3} = \frac{- 2}{3}$

Step 8.2.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 2}{3}$

Step 8.2.2.3

Simplify the right side.

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Step 8.2.2.3.1

Move the negative in front of the fraction.

$a = - \frac{2}{3}$

Step 9

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

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Step 9.1

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

$a + 2 = 0$

Step 9.2

Subtract $2$ from both sides of the equation.

$a = - 2$

Step 10

The final solution is all the values that make $5 a (3 a + 2) (a + 2) = 0$ true.

$a = 0, - \frac{2}{3}, - 2$

Step 11

The result can be shown in multiple forms.

Exact Form:

$a = 0, - \frac{2}{3}, - 2$

Decimal Form:

$a = 0, - 0.\overline{6}, - 2$