Finite Math Examples

$1 + 3 + 5 + (2 n - 1) = n^{2}$

Step 1

Simplify $1 + 3 + 5 + 2 n - 1$ .

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Step 1.1

Combine the opposite terms in $1 + 3 + 5 + 2 n - 1$ .

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Step 1.1.1

Subtract $1$ from $1$ .

$3 + 5 + 2 n + 0 = n^{2}$

Step 1.1.2

Add $3 + 5 + 2 n$ and $0$ .

$3 + 5 + 2 n = n^{2}$

Step 1.2

Add $3$ and $5$ .

$8 + 2 n = n^{2}$

Step 2

Subtract $n^{2}$ from both sides of the equation.

$8 + 2 n - n^{2} = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $- 1$ out of $8 + 2 n - n^{2}$ .

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Step 3.1.1

Reorder the expression.

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Step 3.1.1.1

Move $8$ .

$2 n - n^{2} + 8 = 0$

Step 3.1.1.2

Reorder $2 n$ and $- n^{2}$ .

$- n^{2} + 2 n + 8 = 0$

Step 3.1.2

Factor $- 1$ out of $- n^{2}$ .

$- (n^{2}) + 2 n + 8 = 0$

Step 3.1.3

Factor $- 1$ out of $2 n$ .

$- (n^{2}) - (- 2 n) + 8 = 0$

Step 3.1.4

Rewrite $8$ as $- 1 (- 8)$ .

$- (n^{2}) - (- 2 n) - 1 \cdot - 8 = 0$

Step 3.1.5

Factor $- 1$ out of $- (n^{2}) - (- 2 n)$ .

$- (n^{2} - 2 n) - 1 \cdot - 8 = 0$

Step 3.1.6

Factor $- 1$ out of $- (n^{2} - 2 n) - 1 (- 8)$ .

$- (n^{2} - 2 n - 8) = 0$

Step 3.2

Factor.

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Step 3.2.1

Factor $n^{2} - 2 n - 8$ using the AC method.

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Step 3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 3.2.1.2

Write the factored form using these integers.

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

Step 3.2.2

Remove unnecessary parentheses.

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

Step 4

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

$n - 4 = 0$

$n + 2 = 0$

Step 5

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

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Step 5.1

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

$n - 4 = 0$

Step 5.2

Add $4$ to both sides of the equation.

$n = 4$

Step 6

Set $n + 2$ equal to $0$ and solve for $n$ .

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Step 6.1

Set $n + 2$ equal to $0$ .

$n + 2 = 0$

Step 6.2

Subtract $2$ from both sides of the equation.

$n = - 2$

Step 7

The final solution is all the values that make $- (n - 4) (n + 2) = 0$ true.

$n = 4, - 2$