Algebra Examples

2 (n - 2) (n + 1) - (n + 3) = 0

$2 (n - 2) (n + 1) - (n + 3) = 0$

Step 1

Simplify the left side.

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Step 1.1

Simplify

2 (n - 2) (n + 1) - (n + 3)

$2 (n - 2) (n + 1) - (n + 3)$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Apply the distributive property.

(2 n + 2 \cdot - 2) (n + 1) - (n + 3) = 0

$(2 n + 2 \cdot - 2) (n + 1) - (n + 3) = 0$

Step 1.1.1.2

Multiply

2

$2$ by

- 2

$- 2$ .

(2 n - 4) (n + 1) - (n + 3) = 0

$(2 n - 4) (n + 1) - (n + 3) = 0$

Step 1.1.1.3

Expand

(2 n - 4) (n + 1)

$(2 n - 4) (n + 1)$ using the FOIL Method.

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Step 1.1.1.3.1

Apply the distributive property.

2 n (n + 1) - 4 (n + 1) - (n + 3) = 0

$2 n (n + 1) - 4 (n + 1) - (n + 3) = 0$

Step 1.1.1.3.2

Apply the distributive property.

2 n \cdot n + 2 n \cdot 1 - 4 (n + 1) - (n + 3) = 0

$2 n \cdot n + 2 n \cdot 1 - 4 (n + 1) - (n + 3) = 0$

Step 1.1.1.3.3

Apply the distributive property.

2 n \cdot n + 2 n \cdot 1 - 4 n - 4 \cdot 1 - (n + 3) = 0

$2 n \cdot n + 2 n \cdot 1 - 4 n - 4 \cdot 1 - (n + 3) = 0$

2 n \cdot n + 2 n \cdot 1 - 4 n - 4 \cdot 1 - (n + 3) = 0

$2 n \cdot n + 2 n \cdot 1 - 4 n - 4 \cdot 1 - (n + 3) = 0$

Step 1.1.1.4

Simplify and combine like terms.

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Step 1.1.1.4.1

Simplify each term.

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Step 1.1.1.4.1.1

Multiply

n

$n$ by

n

$n$ by adding the exponents.

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Step 1.1.1.4.1.1.1

Move

n

$n$ .

2 (n \cdot n) + 2 n \cdot 1 - 4 n - 4 \cdot 1 - (n + 3) = 0

$2 (n \cdot n) + 2 n \cdot 1 - 4 n - 4 \cdot 1 - (n + 3) = 0$

Step 1.1.1.4.1.1.2

Multiply

n

$n$ by

n

$n$ .

2 n^{2} + 2 n \cdot 1 - 4 n - 4 \cdot 1 - (n + 3) = 0

$2 n^{2} + 2 n \cdot 1 - 4 n - 4 \cdot 1 - (n + 3) = 0$

2 n^{2} + 2 n \cdot 1 - 4 n - 4 \cdot 1 - (n + 3) = 0

$2 n^{2} + 2 n \cdot 1 - 4 n - 4 \cdot 1 - (n + 3) = 0$

Step 1.1.1.4.1.2

Multiply

2

$2$ by

1

$1$ .

2 n^{2} + 2 n - 4 n - 4 \cdot 1 - (n + 3) = 0

$2 n^{2} + 2 n - 4 n - 4 \cdot 1 - (n + 3) = 0$

Step 1.1.1.4.1.3

Multiply

- 4

$- 4$ by

1

$1$ .

2 n^{2} + 2 n - 4 n - 4 - (n + 3) = 0

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

2 n^{2} + 2 n - 4 n - 4 - (n + 3) = 0

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

Step 1.1.1.4.2

Subtract

4 n

$4 n$ from

2 n

$2 n$ .

2 n^{2} - 2 n - 4 - (n + 3) = 0

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

2 n^{2} - 2 n - 4 - (n + 3) = 0

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

Step 1.1.1.5

Apply the distributive property.

2 n^{2} - 2 n - 4 - n - 1 \cdot 3 = 0

$2 n^{2} - 2 n - 4 - n - 1 \cdot 3 = 0$

Step 1.1.1.6

Multiply

- 1

$- 1$ by

3

$3$ .

2 n^{2} - 2 n - 4 - n - 3 = 0

$2 n^{2} - 2 n - 4 - n - 3 = 0$

2 n^{2} - 2 n - 4 - n - 3 = 0

$2 n^{2} - 2 n - 4 - n - 3 = 0$

Step 1.1.2

Simplify by adding terms.

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Step 1.1.2.1

Subtract

n

$n$ from

- 2 n

$- 2 n$ .

2 n^{2} - 3 n - 4 - 3 = 0

$2 n^{2} - 3 n - 4 - 3 = 0$

Step 1.1.2.2

Subtract

3

$3$ from

- 4

$- 4$ .

2 n^{2} - 3 n - 7 = 0

$2 n^{2} - 3 n - 7 = 0$

2 n^{2} - 3 n - 7 = 0

$2 n^{2} - 3 n - 7 = 0$

2 n^{2} - 3 n - 7 = 0

$2 n^{2} - 3 n - 7 = 0$

2 n^{2} - 3 n - 7 = 0

$2 n^{2} - 3 n - 7 = 0$

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 = 2

$a = 2$ ,

b = - 3

$b = - 3$ , and

c = - 7

$c = - 7$ into the quadratic formula and solve for

n

$n$ .

\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (2 \cdot - 7)}}{2 \cdot 2}

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (2 \cdot - 7)}}{2 \cdot 2}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise

- 3

$- 3$ to the power of

2

$2$ .

n = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 7}}{2 \cdot 2}

$n = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 7}}{2 \cdot 2}$

Step 4.1.2

Multiply

- 4 \cdot 2 \cdot - 7

$- 4 \cdot 2 \cdot - 7$ .

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Step 4.1.2.1

Multiply

- 4

$- 4$ by

2

$2$ .

n = \frac{3 \pm \sqrt{9 - 8 \cdot - 7}}{2 \cdot 2}

$n = \frac{3 \pm \sqrt{9 - 8 \cdot - 7}}{2 \cdot 2}$

Step 4.1.2.2

Multiply

- 8

$- 8$ by

$- 7$ .

$n = \frac{3 \pm \sqrt{9 + 56}}{2 \cdot 2}$

Step 4.1.3

Add

$9$ and

$56$ .

$n = \frac{3 \pm \sqrt{65}}{2 \cdot 2}$

Step 4.2

Multiply

$2$ by

$2$ .

$n = \frac{3 \pm \sqrt{65}}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$n = \frac{3 \pm \sqrt{65}}{4}$

Decimal Form:

$n = 2.76556443 \dots, - 1.26556443 \dots$