Precalculus Examples

$4 v^{2} - 3 = 11 v$

Step 1

Subtract $11 v$ from both sides of the equation.

$4 v^{2} - 3 - 11 v = 0$

Step 2

Factor by grouping.

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Step 2.1

Reorder terms.

$4 v^{2} - 11 v - 3 = 0$

Step 2.2

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 $a \cdot c = 4 \cdot - 3 = - 12$ and whose sum is $b = - 11$ .

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Step 2.2.1

Factor $- 11$ out of $- 11 v$ .

$4 v^{2} - 11 v - 3 = 0$

Step 2.2.2

Rewrite $- 11$ as $1$ plus $- 12$

$4 v^{2} + (1 - 12) v - 3 = 0$

Step 2.2.3

Apply the distributive property.

$4 v^{2} + 1 v - 12 v - 3 = 0$

Step 2.2.4

Multiply $v$ by $1$ .

$4 v^{2} + v - 12 v - 3 = 0$

Step 2.3

Factor out the greatest common factor from each group.

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Step 2.3.1

Group the first two terms and the last two terms.

$(4 v^{2} + v) - 12 v - 3 = 0$

Step 2.3.2

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

$v (4 v + 1) - 3 (4 v + 1) = 0$

Step 2.4

Factor the polynomial by factoring out the greatest common factor, $4 v + 1$ .

$(4 v + 1) (v - 3) = 0$

Step 3

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

$4 v + 1 = 0$

$v - 3 = 0$

Step 4

Set $4 v + 1$ equal to $0$ and solve for $v$ .

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Step 4.1

Set $4 v + 1$ equal to $0$ .

$4 v + 1 = 0$

Step 4.2

Solve $4 v + 1 = 0$ for $v$ .

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$4 v = - 1$

Step 4.2.2

Divide each term in $4 v = - 1$ by $4$ and simplify.

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Step 4.2.2.1

Divide each term in $4 v = - 1$ by $4$ .

$\frac{4 v}{4} = \frac{- 1}{4}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{4 v}{4} = \frac{- 1}{4}$

Step 4.2.2.2.1.2

Divide $v$ by $1$ .

$v = \frac{- 1}{4}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Move the negative in front of the fraction.

$v = - \frac{1}{4}$

Step 5

Set $v - 3$ equal to $0$ and solve for $v$ .

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Step 5.1

Set $v - 3$ equal to $0$ .

$v - 3 = 0$

Step 5.2

Add $3$ to both sides of the equation.

$v = 3$

Step 6

The final solution is all the values that make $(4 v + 1) (v - 3) = 0$ true.

$v = - \frac{1}{4}, 3$

Step 7

The result can be shown in multiple forms.

Exact Form:

$v = - \frac{1}{4}, 3$

Decimal Form:

$v = - 0.25, 3$