Solve 32n/24n=4n^2/3n

$$\frac { 32 n } { 24 n } = \frac { 4 n ^ { 2 } } { 3 n }$$

Solve for n

$$n=1$$

Steps Using Factoring

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 24n, the least common multiple of 24n,3n.

$$32n=8\times 4n^{2}$$

Multiply 8 and 4 to get 32.

$$32n=32n^{2}$$

Subtract 32n^{2} from both sides.

$$32n-32n^{2}=0$$

Factor out n.

$$n\left(32-32n\right)=0$$

To find equation solutions, solve n=0 and 32-32n=0.

$$n=0 n=1$$

Variable n cannot be equal to 0.

$$n=1$$

Steps Using the Quadratic Formula

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 24n, the least common multiple of 24n,3n.

$$32n=8\times 4n^{2}$$

Multiply 8 and 4 to get 32.

$$32n=32n^{2}$$

Subtract 32n^{2} from both sides.

$$32n-32n^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-32n^{2}+32n=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -32 for a, 32 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-32±\sqrt{32^{2}}}{2\left(-32\right)}$$

Take the square root of 32^{2}.

$$n=\frac{-32±32}{2\left(-32\right)}$$

Multiply 2 times -32.

$$n=\frac{-32±32}{-64}$$

Now solve the equation n=\frac{-32±32}{-64} when ± is plus. Add -32 to 32.

$$n=\frac{0}{-64}$$

Divide 0 by -64.

$$n=0$$

Now solve the equation n=\frac{-32±32}{-64} when ± is minus. Subtract 32 from -32.

$$n=-\frac{64}{-64}$$

Divide -64 by -64.

$$n=1$$

The equation is now solved.

$$n=0 n=1$$

Variable n cannot be equal to 0.

$$n=1$$

Steps for Completing the Square

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 24n, the least common multiple of 24n,3n.

$$32n=8\times 4n^{2}$$

Multiply 8 and 4 to get 32.

$$32n=32n^{2}$$

Subtract 32n^{2} from both sides.

$$32n-32n^{2}=0$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-32n^{2}+32n=0$$

Divide both sides by -32.

$$\frac{-32n^{2}+32n}{-32}=\frac{0}{-32}$$

Dividing by -32 undoes the multiplication by -32.

$$n^{2}+\frac{32}{-32}n=\frac{0}{-32}$$

Divide 32 by -32.

$$n^{2}-n=\frac{0}{-32}$$

Divide 0 by -32.

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

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-n+\left(-\frac{1}{2}\right)^{2}=\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-n+\frac{1}{4}=\frac{1}{4}$$

Factor n^{2}-n+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{1}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{1}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$n-\frac{1}{2}=\frac{1}{2} n-\frac{1}{2}=-\frac{1}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$n=1 n=0$$

Variable n cannot be equal to 0.

$$n=1$$