Trigonometry Examples

$4 v^{2} = - 8 v - 3$

Step 1

Add $8 v$ to both sides of the equation.

$4 v^{2} + 8 v = - 3$

Step 2

Add $3$ to both sides of the equation.

$4 v^{2} + 8 v + 3 = 0$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 4 \cdot 3 = 12$ and whose sum is $b = 8$ .

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Step 3.1.1

Factor $8$ out of $8 v$ .

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

Step 3.1.2

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

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

Step 3.1.3

Apply the distributive property.

$4 v^{2} + 2 v + 6 v + 3 = 0$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

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

Step 3.2.2

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

$2 v (2 v + 1) + 3 (2 v + 1) = 0$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $2 v + 1$ .

$(2 v + 1) (2 v + 3) = 0$

Step 4

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

$2 v + 1 = 0$

$2 v + 3 = 0$

Step 5

Set $2 v + 1$ equal to $0$ and solve for $v$ .

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Step 5.1

Set $2 v + 1$ equal to $0$ .

$2 v + 1 = 0$

Step 5.2

Solve $2 v + 1 = 0$ for $v$ .

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$2 v = - 1$

Step 5.2.2

Divide each term in $2 v = - 1$ by $2$ and simplify.

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Step 5.2.2.1

Divide each term in $2 v = - 1$ by $2$ .

$\frac{2 v}{2} = \frac{- 1}{2}$

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.

$\frac{2 v}{2} = \frac{- 1}{2}$

Step 5.2.2.2.1.2

Divide $v$ by $1$ .

$v = \frac{- 1}{2}$

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.

$v = - \frac{1}{2}$

Step 6

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

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Step 6.1

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

$2 v + 3 = 0$

Step 6.2

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

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Step 6.2.1

Subtract $3$ from both sides of the equation.

$2 v = - 3$

Step 6.2.2

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

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Step 6.2.2.1

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

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

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.2.1.2

Divide $v$ by $1$ .

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

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Move the negative in front of the fraction.

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

Step 7

The final solution is all the values that make $(2 v + 1) (2 v + 3) = 0$ true.

$v = - \frac{1}{2}, - \frac{3}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$v = - \frac{1}{2}, - \frac{3}{2}$

Decimal Form:

$v = - 0.5, - 1.5$

Mixed Number Form:

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