Solve by Factoring (4n-37)/3=10/n

$\frac{4 n - 37}{3} = \frac{10}{n}$

Step 1

Subtract $\frac{10}{n}$ from both sides of the equation.

$\frac{4 n - 37}{3} - \frac{10}{n} = 0$

Step 2

Simplify $\frac{4 n - 37}{3} - \frac{10}{n}$ .

To write $\frac{4 n - 37}{3}$ as a fraction with a common denominator, multiply by $\frac{n}{n}$ .

$\frac{4 n - 37}{3} \cdot \frac{n}{n} - \frac{10}{n} = 0$

To write $- \frac{10}{n}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{4 n - 37}{3} \cdot \frac{n}{n} - \frac{10}{n} \cdot \frac{3}{3} = 0$

Write each expression with a common denominator of $3 n$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{4 n - 37}{3}$ by $\frac{n}{n}$ .

$\frac{(4 n - 37) n}{3 n} - \frac{10}{n} \cdot \frac{3}{3} = 0$

Multiply $\frac{10}{n}$ by $\frac{3}{3}$ .

$\frac{(4 n - 37) n}{3 n} - \frac{10 \cdot 3}{n \cdot 3} = 0$

Reorder the factors of $n \cdot 3$ .

$\frac{(4 n - 37) n}{3 n} - \frac{10 \cdot 3}{3 n} = 0$

Combine the numerators over the common denominator.

$\frac{(4 n - 37) n - 10 \cdot 3}{3 n} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{4 n \cdot n - 37 n - 10 \cdot 3}{3 n} = 0$

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

Move $n$ .

$\frac{4 (n \cdot n) - 37 n - 10 \cdot 3}{3 n} = 0$

Multiply $n$ by $n$ .

$\frac{4 n^{2} - 37 n - 10 \cdot 3}{3 n} = 0$

Multiply $- 10$ by $3$ .

$\frac{4 n^{2} - 37 n - 30}{3 n} = 0$

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 $a \cdot c = 4 \cdot - 30 = - 120$ and whose sum is $b = - 37$ .

Factor $- 37$ out of $- 37 n$ .

$\frac{4 n^{2} - 37 (n) - 30}{3 n} = 0$

Rewrite $- 37$ as $3$ plus $- 40$

$\frac{4 n^{2} + (3 - 40) n - 30}{3 n} = 0$

Apply the distributive property.

$\frac{4 n^{2} + 3 n - 40 n - 30}{3 n} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(4 n^{2} + 3 n) - 40 n - 30}{3 n} = 0$

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

$\frac{n (4 n + 3) - 10 (4 n + 3)}{3 n} = 0$

Factor the polynomial by factoring out the greatest common factor, $4 n + 3$ .

$\frac{(4 n + 3) (n - 10)}{3 n} = 0$

Step 3

Set the numerator equal to zero.

$(4 n + 3) (n - 10) = 0$

Step 4

Solve the equation for $n$ .

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

$4 n + 3 = 0$

$n - 10 = 0$

Set $4 n + 3$ equal to $0$ and solve for $n$ .

Set $4 n + 3$ equal to $0$ .

$4 n + 3 = 0$

Solve $4 n + 3 = 0$ for $n$ .

Subtract $3$ from both sides of the equation.

$4 n = - 3$

Divide each term in $4 n = - 3$ by $4$ and simplify.

Divide each term in $4 n = - 3$ by $4$ .

$\frac{4 n}{4} = \frac{- 3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 n}{4} = \frac{- 3}{4}$

Divide $n$ by $1$ .

$n = \frac{- 3}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$n = - \frac{3}{4}$

Set $n - 10$ equal to $0$ and solve for $n$ .

Set $n - 10$ equal to $0$ .

$n - 10 = 0$

Add $10$ to both sides of the equation.

$n = 10$

The final solution is all the values that make $(4 n + 3) (n - 10) = 0$ true.

$n = - \frac{3}{4}, 10$

Solve by Factoring (4n-37)/3=10/n

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