Solve for n 9n^2-23n+20=3n^2

$9 n^{2} - 23 n + 20 = 3 n^{2}$

Step 1

Move all terms containing $n$ to the left side of the equation.

Subtract $3 n^{2}$ from both sides of the equation.

$9 n^{2} - 23 n + 20 - 3 n^{2} = 0$

Subtract $3 n^{2}$ from $9 n^{2}$ .

$6 n^{2} - 23 n + 20 = 0$

Step 2

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 $a \cdot c = 6 \cdot 20 = 120$ and whose sum is $b = - 23$ .

Factor $- 23$ out of $- 23 n$ .

$6 n^{2} - 23 n + 20 = 0$

Rewrite $- 23$ as $- 8$ plus $- 15$

$6 n^{2} + (- 8 - 15) n + 20 = 0$

Apply the distributive property.

$6 n^{2} - 8 n - 15 n + 20 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(6 n^{2} - 8 n) - 15 n + 20 = 0$

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

$2 n (3 n - 4) - 5 (3 n - 4) = 0$

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

$(3 n - 4) (2 n - 5) = 0$

Step 3

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

$3 n - 4 = 0$

$2 n - 5 = 0$

Step 4

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

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

$3 n - 4 = 0$

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

Add $4$ to both sides of the equation.

$3 n = 4$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $n$ by $1$ .

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

Step 5

Set $2 n - 5$ equal to $0$ and solve for $n$ .

Set $2 n - 5$ equal to $0$ .

$2 n - 5 = 0$

Solve $2 n - 5 = 0$ for $n$ .

Add $5$ to both sides of the equation.

$2 n = 5$

Divide each term in $2 n = 5$ by $2$ and simplify.

Divide each term in $2 n = 5$ by $2$ .

$\frac{2 n}{2} = \frac{5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 n}{2} = \frac{5}{2}$

Divide $n$ by $1$ .

$n = \frac{5}{2}$

Step 6

The final solution is all the values that make $(3 n - 4) (2 n - 5) = 0$ true.

$n = \frac{4}{3}, \frac{5}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$n = \frac{4}{3}, \frac{5}{2}$

Decimal Form:

$n = 1.\overline{3}, 2.5$

Mixed Number Form:

$n = 1 \frac{1}{3}, 2 \frac{1}{2}$

Solve for n 9n^2-23n+20=3n^2

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