Algebra Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Apply the distributive property.

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

Step 1.1.1.2

Multiply $2$ by $- 2$ .

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

Apply the distributive property.

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

Step 1.1.1.3.2

Apply the distributive property.

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

Step 1.1.1.3.3

Apply the distributive property.

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

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 $n$ by $n$ by adding the exponents.

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Step 1.1.1.4.1.1.1

Move $n$ .

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

Step 1.1.1.4.1.1.2

Multiply $n$ by $n$ .

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

Step 1.1.1.4.1.2

Multiply $2$ by $1$ .

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

Step 1.1.1.4.1.3

Multiply $- 4$ by $1$ .

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

Step 1.1.1.4.2

Subtract $4 n$ from $2 n$ .

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

Step 1.1.1.5

Apply the distributive property.

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

Step 1.1.1.6

Multiply $- 1$ by $3$ .

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

Step 1.1.2

Simplify by adding terms.

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Step 1.1.2.1

Subtract $n$ from $- 2 n$ .

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

Step 1.1.2.2

Subtract $3$ from $- 4$ .

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

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 2$ , $b = - 3$ , and $c = - 7$ into the quadratic formula and solve for $n$ .

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 3$ to the power of $2$ .

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

Step 4.1.2

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

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

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

Step 4.1.2.2

Multiply $- 8$ 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:

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