Finite Math Examples

$- 42 = - 13 n + n^{2}$

Step 1

Rewrite the equation as $- 13 n + n^{2} = - 42$ .

$- 13 n + n^{2} = - 42$

Step 2

Add $42$ to both sides of the equation.

$- 13 n + n^{2} + 42 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Let $u = n$ . Substitute $u$ for all occurrences of $n$ .

$- 13 u + u^{2} + 42 = 0$

Step 3.2

Factor $- 13 u + u^{2} + 42$ using the AC method.

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Step 3.2.1

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 $42$ and whose sum is $- 13$ .

$- 7, - 6$

Step 3.2.2

Write the factored form using these integers.

$(u - 7) (u - 6) = 0$

Step 3.3

Replace all occurrences of $u$ with $n$ .

$(n - 7) (n - 6) = 0$

Step 4

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

$n - 7 = 0$

$n - 6 = 0$

Step 5

Set $n - 7$ equal to $0$ and solve for $n$ .

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Step 5.1

Set $n - 7$ equal to $0$ .

$n - 7 = 0$

Step 5.2

Add $7$ to both sides of the equation.

$n = 7$

Step 6

Set $n - 6$ equal to $0$ and solve for $n$ .

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Step 6.1

Set $n - 6$ equal to $0$ .

$n - 6 = 0$

Step 6.2

Add $6$ to both sides of the equation.

$n = 6$

Step 7

The final solution is all the values that make $(n - 7) (n - 6) = 0$ true.

$n = 7, 6$