Solve by Factoring 16n^2-114n=-14

$16 n^{2} - 114 n = - 14$

Step 1

Add $14$ to both sides of the equation.

$16 n^{2} - 114 n + 14 = 0$

Step 2

Factor $2$ out of $16 n^{2} - 114 n + 14$ .

Factor $2$ out of $16 n^{2}$ .

$2 (8 n^{2}) - 114 n + 14 = 0$

Factor $2$ out of $- 114 n$ .

$2 (8 n^{2}) + 2 (- 57 n) + 14 = 0$

Factor $2$ out of $14$ .

$2 (8 n^{2}) + 2 (- 57 n) + 2 (7) = 0$

Factor $2$ out of $2 (8 n^{2}) + 2 (- 57 n)$ .

$2 (8 n^{2} - 57 n) + 2 (7) = 0$

Factor $2$ out of $2 (8 n^{2} - 57 n) + 2 (7)$ .

$2 (8 n^{2} - 57 n + 7) = 0$

Step 3

Factor.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 8 \cdot 7 = 56$ and whose sum is $b = - 57$ .

Factor $- 57$ out of $- 57 n$ .

$2 (8 n^{2} - 57 n + 7) = 0$

Rewrite $- 57$ as $- 1$ plus $- 56$

$2 (8 n^{2} + (- 1 - 56) n + 7) = 0$

Apply the distributive property.

$2 (8 n^{2} - 1 n - 56 n + 7) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$2 ((8 n^{2} - 1 n) - 56 n + 7) = 0$

Factor out the greatest common factor (GCF) from each group.

$2 (n (8 n - 1) - 7 (8 n - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $8 n - 1$ .

$2 ((8 n - 1) (n - 7)) = 0$

Remove unnecessary parentheses.

$2 (8 n - 1) (n - 7) = 0$

Step 4

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

$8 n - 1 = 0$

$n - 7 = 0$

Step 5

Set $8 n - 1$ equal to $0$ and solve for $n$ .

Set $8 n - 1$ equal to $0$ .

$8 n - 1 = 0$

Solve $8 n - 1 = 0$ for $n$ .

Add $1$ to both sides of the equation.

$8 n = 1$

Divide each term in $8 n = 1$ by $8$ and simplify.

Divide each term in $8 n = 1$ by $8$ .

$\frac{8 n}{8} = \frac{1}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 n}{8} = \frac{1}{8}$

Divide $n$ by $1$ .

$n = \frac{1}{8}$

Step 6

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

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

$n - 7 = 0$

Add $7$ to both sides of the equation.

$n = 7$

Step 7

The final solution is all the values that make $2 (8 n - 1) (n - 7) = 0$ true.

$n = \frac{1}{8}, 7$

Solve by Factoring 16n^2-114n=-14

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