Trigonometry Examples

$(3 - y) (y + 4) = 3 y - 5$

Step 1

Simplify $(3 - y) (y + 4)$ .

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Step 1.1

Rewrite.

$0 + 0 + (3 - y) (y + 4) = 3 y - 5$

Step 1.2

Simplify by adding zeros.

$(3 - y) (y + 4) = 3 y - 5$

Step 1.3

Expand $(3 - y) (y + 4)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$3 (y + 4) - y (y + 4) = 3 y - 5$

Step 1.3.2

Apply the distributive property.

$3 y + 3 \cdot 4 - y (y + 4) = 3 y - 5$

Step 1.3.3

Apply the distributive property.

$3 y + 3 \cdot 4 - y \cdot y - y \cdot 4 = 3 y - 5$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $3$ by $4$ .

$3 y + 12 - y \cdot y - y \cdot 4 = 3 y - 5$

Step 1.4.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 1.4.1.2.1

Move $y$ .

$3 y + 12 - (y \cdot y) - y \cdot 4 = 3 y - 5$

Step 1.4.1.2.2

Multiply $y$ by $y$ .

$3 y + 12 - y^{2} - y \cdot 4 = 3 y - 5$

Step 1.4.1.3

Multiply $4$ by $- 1$ .

$3 y + 12 - y^{2} - 4 y = 3 y - 5$

Step 1.4.2

Subtract $4 y$ from $3 y$ .

$- y + 12 - y^{2} = 3 y - 5$

Step 2

Move all terms containing $y$ to the left side of the equation.

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Step 2.1

Subtract $3 y$ from both sides of the equation.

$- y + 12 - y^{2} - 3 y = - 5$

Step 2.2

Subtract $3 y$ from $- y$ .

$- 4 y + 12 - y^{2} = - 5$

Step 3

Move all terms to the left side of the equation and simplify.

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Step 3.1

Add $5$ to both sides of the equation.

$- 4 y + 12 - y^{2} + 5 = 0$

Step 3.2

Add $12$ and $5$ .

$- 4 y - y^{2} + 17 = 0$

Step 4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5

Substitute the values $a = - 1$ , $b = - 4$ , and $c = 17$ into the quadratic formula and solve for $y$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (- 1 \cdot 17)}}{2 \cdot - 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 4$ to the power of $2$ .

$y = \frac{4 \pm \sqrt{16 - 4 \cdot - 1 \cdot 17}}{2 \cdot - 1}$

Step 6.1.2

Multiply $- 4 \cdot - 1 \cdot 17$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 1$ .

$y = \frac{4 \pm \sqrt{16 + 4 \cdot 17}}{2 \cdot - 1}$

Step 6.1.2.2

Multiply $4$ by $17$ .

$y = \frac{4 \pm \sqrt{16 + 68}}{2 \cdot - 1}$

Step 6.1.3

Add $16$ and $68$ .

$y = \frac{4 \pm \sqrt{84}}{2 \cdot - 1}$

Step 6.1.4

Rewrite $84$ as $2^{2} \cdot 21$ .

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Step 6.1.4.1

Factor $4$ out of $84$ .

$y = \frac{4 \pm \sqrt{4 (21)}}{2 \cdot - 1}$

Step 6.1.4.2

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

$y = \frac{4 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot - 1}$

Step 6.1.5

Pull terms out from under the radical.

$y = \frac{4 \pm 2 \sqrt{21}}{2 \cdot - 1}$

Step 6.2

Multiply $2$ by $- 1$ .

$y = \frac{4 \pm 2 \sqrt{21}}{- 2}$

Step 6.3

Simplify $\frac{4 \pm 2 \sqrt{21}}{- 2}$ .

$y = \frac{2 \pm \sqrt{21}}{- 1}$

Step 6.4

Move the negative one from the denominator of $\frac{2 \pm \sqrt{21}}{- 1}$ .

$y = - 1 \cdot (2 \pm \sqrt{21})$

Step 6.5

Rewrite $- 1 \cdot (2 \pm \sqrt{21})$ as $- (2 \pm \sqrt{21})$ .

$y = - (2 \pm \sqrt{21})$

Step 7

The final answer is the combination of both solutions.

$y = - 2 - \sqrt{21}, - 2 + \sqrt{21}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$y = - 2 - \sqrt{21}, - 2 + \sqrt{21}$

Decimal Form:

$y = - 6.58257569 \dots, 2.58257569 \dots$