Solve 6500=n[595-15n)

$$6500 = n [ 595 - 15 n )$$

Solve for n

$$n=\frac{119+\sqrt{1439}i}{6}\approx 19.833333333+6.322358913i$$

$$n=\frac{-\sqrt{1439}i+119}{6}\approx 19.833333333-6.322358913i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply n by 595-15n.

$$6500=595n-15n^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$595n-15n^{2}=6500$$

Subtract 6500 from both sides.

$$595n-15n^{2}-6500=0$$

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.

$$-15n^{2}+595n-6500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -15 for a, 595 for b, and -6500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-595±\sqrt{595^{2}-4\left(-15\right)\left(-6500\right)}}{2\left(-15\right)}$$

Square 595.

$$n=\frac{-595±\sqrt{354025-4\left(-15\right)\left(-6500\right)}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$n=\frac{-595±\sqrt{354025+60\left(-6500\right)}}{2\left(-15\right)}$$

Multiply 60 times -6500.

$$n=\frac{-595±\sqrt{354025-390000}}{2\left(-15\right)}$$

Add 354025 to -390000.

$$n=\frac{-595±\sqrt{-35975}}{2\left(-15\right)}$$

Take the square root of -35975.

$$n=\frac{-595±5\sqrt{1439}i}{2\left(-15\right)}$$

Multiply 2 times -15.

$$n=\frac{-595±5\sqrt{1439}i}{-30}$$

Now solve the equation n=\frac{-595±5\sqrt{1439}i}{-30} when ± is plus. Add -595 to 5i\sqrt{1439}.

$$n=\frac{-595+5\sqrt{1439}i}{-30}$$

Divide -595+5i\sqrt{1439} by -30.

$$n=\frac{-\sqrt{1439}i+119}{6}$$

Now solve the equation n=\frac{-595±5\sqrt{1439}i}{-30} when ± is minus. Subtract 5i\sqrt{1439} from -595.

$$n=\frac{-5\sqrt{1439}i-595}{-30}$$

Divide -595-5i\sqrt{1439} by -30.

$$n=\frac{119+\sqrt{1439}i}{6}$$

The equation is now solved.

$$n=\frac{-\sqrt{1439}i+119}{6} n=\frac{119+\sqrt{1439}i}{6}$$

Steps for Completing the Square

Use the distributive property to multiply n by 595-15n.

$$6500=595n-15n^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$595n-15n^{2}=6500$$

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.

$$-15n^{2}+595n=6500$$

Divide both sides by -15.

$$\frac{-15n^{2}+595n}{-15}=\frac{6500}{-15}$$

Dividing by -15 undoes the multiplication by -15.

$$n^{2}+\frac{595}{-15}n=\frac{6500}{-15}$$

Reduce the fraction \frac{595}{-15} to lowest terms by extracting and canceling out 5.

$$n^{2}-\frac{119}{3}n=\frac{6500}{-15}$$

Reduce the fraction \frac{6500}{-15} to lowest terms by extracting and canceling out 5.

$$n^{2}-\frac{119}{3}n=-\frac{1300}{3}$$

Divide -\frac{119}{3}, the coefficient of the x term, by 2 to get -\frac{119}{6}. Then add the square of -\frac{119}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{119}{3}n+\left(-\frac{119}{6}\right)^{2}=-\frac{1300}{3}+\left(-\frac{119}{6}\right)^{2}$$

Square -\frac{119}{6} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{119}{3}n+\frac{14161}{36}=-\frac{1300}{3}+\frac{14161}{36}$$

Add -\frac{1300}{3} to \frac{14161}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-\frac{119}{3}n+\frac{14161}{36}=-\frac{1439}{36}$$

Factor n^{2}-\frac{119}{3}n+\frac{14161}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{119}{6}\right)^{2}=-\frac{1439}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{119}{6}\right)^{2}}=\sqrt{-\frac{1439}{36}}$$

Simplify.

$$n-\frac{119}{6}=\frac{\sqrt{1439}i}{6} n-\frac{119}{6}=-\frac{\sqrt{1439}i}{6}$$

Add \frac{119}{6} to both sides of the equation.

$$n=\frac{119+\sqrt{1439}i}{6} n=\frac{-\sqrt{1439}i+119}{6}$$