Trigonometry Examples

$4 w^{2} = 15 w + 4$

Step 1

Subtract $15 w$ from both sides of the equation.

$4 w^{2} - 15 w = 4$

Step 2

Subtract $4$ from both sides of the equation.

$4 w^{2} - 15 w - 4 = 0$

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 $a \cdot c = 4 \cdot - 4 = - 16$ and whose sum is $b = - 15$ .

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Step 3.1.1

Factor $- 15$ out of $- 15 w$ .

$4 w^{2} - 15 w - 4 = 0$

Step 3.1.2

Rewrite $- 15$ as $1$ plus $- 16$

$4 w^{2} + (1 - 16) w - 4 = 0$

Step 3.1.3

Apply the distributive property.

$4 w^{2} + 1 w - 16 w - 4 = 0$

Step 3.1.4

Multiply $w$ by $1$ .

$4 w^{2} + w - 16 w - 4 = 0$

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.

$(4 w^{2} + w) - 16 w - 4 = 0$

Step 3.2.2

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

$w (4 w + 1) - 4 (4 w + 1) = 0$

Step 3.3

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

$(4 w + 1) (w - 4) = 0$

Step 4

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

$4 w + 1 = 0$

$w - 4 = 0$

Step 5

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

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Step 5.1

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

$4 w + 1 = 0$

Step 5.2

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

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$4 w = - 1$

Step 5.2.2

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

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Step 5.2.2.1

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

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

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $4$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.2.1.2

Divide $w$ by $1$ .

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

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.

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

Step 6

Set $w - 4$ equal to $0$ and solve for $w$ .

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Step 6.1

Set $w - 4$ equal to $0$ .

$w - 4 = 0$

Step 6.2

Add $4$ to both sides of the equation.

$w = 4$

Step 7

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

$w = - \frac{1}{4}, 4$

Step 8

The result can be shown in multiple forms.

Exact Form:

$w = - \frac{1}{4}, 4$

Decimal Form:

$w = - 0.25, 4$