Solve n(-5)^4divn^2=5

$$n ( - 5 ) ^ { 4 } \div n ^ { 2 } = 5$$

Solve for n

$$n=125$$

Steps Using Factoring

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by n^{2}.

$$n\left(-5\right)^{4}=5n^{2}$$

Calculate -5 to the power of 4 and get 625.

$$n\times 625=5n^{2}$$

Subtract 5n^{2} from both sides.

$$n\times 625-5n^{2}=0$$

Factor out n.

$$n\left(625-5n\right)=0$$

To find equation solutions, solve n=0 and 625-5n=0.

$$n=0 n=125$$

Variable n cannot be equal to 0.

$$n=125$$

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 n^{2}.

$$n\left(-5\right)^{4}=5n^{2}$$

Calculate -5 to the power of 4 and get 625.

$$n\times 625=5n^{2}$$

Subtract 5n^{2} from both sides.

$$n\times 625-5n^{2}=0$$

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.

$$-5n^{2}+625n=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 625 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-625±\sqrt{625^{2}}}{2\left(-5\right)}$$

Take the square root of 625^{2}.

$$n=\frac{-625±625}{2\left(-5\right)}$$

Multiply 2 times -5.

$$n=\frac{-625±625}{-10}$$

Now solve the equation n=\frac{-625±625}{-10} when ± is plus. Add -625 to 625.

$$n=\frac{0}{-10}$$

Divide 0 by -10.

$$n=0$$

Now solve the equation n=\frac{-625±625}{-10} when ± is minus. Subtract 625 from -625.

$$n=-\frac{1250}{-10}$$

Divide -1250 by -10.

$$n=125$$

The equation is now solved.

$$n=0 n=125$$

Variable n cannot be equal to 0.

$$n=125$$

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 n^{2}.

$$n\left(-5\right)^{4}=5n^{2}$$

Calculate -5 to the power of 4 and get 625.

$$n\times 625=5n^{2}$$

Subtract 5n^{2} from both sides.

$$n\times 625-5n^{2}=0$$

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.

$$-5n^{2}+625n=0$$

Divide both sides by -5.

$$\frac{-5n^{2}+625n}{-5}=\frac{0}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$n^{2}+\frac{625}{-5}n=\frac{0}{-5}$$

Divide 625 by -5.

$$n^{2}-125n=\frac{0}{-5}$$

Divide 0 by -5.

$$n^{2}-125n=0$$

Divide -125, the coefficient of the x term, by 2 to get -\frac{125}{2}. Then add the square of -\frac{125}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-125n+\left(-\frac{125}{2}\right)^{2}=\left(-\frac{125}{2}\right)^{2}$$

Square -\frac{125}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-125n+\frac{15625}{4}=\frac{15625}{4}$$

Factor n^{2}-125n+\frac{15625}{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}.

$$\left(n-\frac{125}{2}\right)^{2}=\frac{15625}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{125}{2}\right)^{2}}=\sqrt{\frac{15625}{4}}$$

Simplify.

$$n-\frac{125}{2}=\frac{125}{2} n-\frac{125}{2}=-\frac{125}{2}$$

Add \frac{125}{2} to both sides of the equation.

$$n=125 n=0$$

Variable n cannot be equal to 0.

$$n=125$$