Algebra Examples

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

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

Step 1

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

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify

(3 - y) (y + 4)

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

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Step 1.1.1.1

Expand

(3 - y) (y + 4)

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

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Step 1.1.1.1.1

Apply the distributive property.

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

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

Step 1.1.1.1.2

Apply the distributive property.

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

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

Step 1.1.1.1.3

Apply the distributive property.

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

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

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

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

Step 1.1.1.2

Simplify and combine like terms.

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Step 1.1.1.2.1

Simplify each term.

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Step 1.1.1.2.1.1

Multiply

3

$3$ by

4

$4$ .

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

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

Step 1.1.1.2.1.2

Multiply

y

$y$ by

y

$y$ by adding the exponents.

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Step 1.1.1.2.1.2.1

Move

y

$y$ .

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

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

Step 1.1.1.2.1.2.2

Multiply

y

$y$ by

y

$y$ .

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

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

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

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

Step 1.1.1.2.1.3

Multiply

4

$4$ by

- 1

$- 1$ .

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

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

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

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

Step 1.1.1.2.2

Subtract

4 y

$4 y$ from

3 y

$3 y$ .

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

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

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

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

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

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

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

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

Step 1.2

Move all the expressions to the left side of the equation.

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Step 1.2.1

Subtract

3 y

$3 y$ from both sides of the equation.

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

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

Step 1.2.2

Add

5

$5$ to both sides of the equation.

- y + 12 - y^{2} - 3 y + 5 = 0

$- y + 12 - y^{2} - 3 y + 5 = 0$

- y + 12 - y^{2} - 3 y + 5 = 0

$- y + 12 - y^{2} - 3 y + 5 = 0$

Step 1.3

Simplify

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

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

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Step 1.3.1

Subtract

3 y

$3 y$ from

- y

$- y$ .

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

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

Step 1.3.2

Add

12

$12$ and

5

$5$ .

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

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

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

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

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

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

Step 2

Use the quadratic formula to find the solutions.

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

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

Step 3

Substitute the values

a = - 1

$a = - 1$ ,

b = - 4

$b = - 4$ , and

c = 17

$c = 17$ into the quadratic formula and solve for

y

$y$ .

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

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise

- 4

$- 4$ to the power of

2

$2$ .

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

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

Step 4.1.2

Multiply

- 4 \cdot - 1 \cdot 17

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

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Step 4.1.2.1

Multiply

- 4

$- 4$ by

- 1

$- 1$ .

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

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

Step 4.1.2.2

Multiply

4

$4$ by

17

$17$ .

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

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

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

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

Step 4.1.3

Add

16

$16$ and

68

$68$ .

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

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

Step 4.1.4

Rewrite

84

$84$ as

2^{2} \cdot 21

$2^{2} \cdot 21$ .

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Step 4.1.4.1

Factor

4

$4$ out of

84

$84$ .

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

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

Step 4.1.4.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

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

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

Step 4.1.5

Pull terms out from under the radical.

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

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

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

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

Step 4.2

Multiply

2

$2$ by

- 1

$- 1$ .

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

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

Step 4.3

Simplify

\frac{4 \pm 2 \sqrt{21}}{- 2}

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

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

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

Step 4.4

Move the negative one from the denominator of

\frac{2 \pm \sqrt{21}}{- 1}

$\frac{2 \pm \sqrt{21}}{- 1}$ .

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

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

Step 4.5

Rewrite

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

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

- (2 \pm \sqrt{21})

$- (2 \pm \sqrt{21})$ .

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

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

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

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

Decimal Form:

y = - 6.58257569 \dots, 2.58257569 \dots

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