Solve -2/n-3=n+9/n+21

$$\frac{ -2 }{ n-3 } = \frac{ n+9 }{ n+21 }$$

Solve for n

$$n=-5$$

$$n=-3$$

Steps Using the Quadratic Formula

Variable n cannot be equal to any of the values -21,3 since division by zero is not defined. Multiply both sides of the equation by \left(n-3\right)\left(n+21\right), the least common multiple of n-3,n+21.

$$\left(n+21\right)\left(-2\right)=\left(n-3\right)\left(n+9\right)$$

Use the distributive property to multiply n+21 by -2.

$$-2n-42=\left(n-3\right)\left(n+9\right)$$

Use the distributive property to multiply n-3 by n+9 and combine like terms.

$$-2n-42=n^{2}+6n-27$$

Subtract n^{2} from both sides.

$$-2n-42-n^{2}=6n-27$$

Subtract 6n from both sides.

$$-2n-42-n^{2}-6n=-27$$

Combine -2n and -6n to get -8n.

$$-8n-42-n^{2}=-27$$

Add 27 to both sides.

$$-8n-42-n^{2}+27=0$$

Add -42 and 27 to get -15.

$$-8n-15-n^{2}=0$$

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.

$$-n^{2}-8n-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -8 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-1\right)\left(-15\right)}}{2\left(-1\right)}$$

Square -8.

$$n=\frac{-\left(-8\right)±\sqrt{64-4\left(-1\right)\left(-15\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$n=\frac{-\left(-8\right)±\sqrt{64+4\left(-15\right)}}{2\left(-1\right)}$$

Multiply 4 times -15.

$$n=\frac{-\left(-8\right)±\sqrt{64-60}}{2\left(-1\right)}$$

Add 64 to -60.

$$n=\frac{-\left(-8\right)±\sqrt{4}}{2\left(-1\right)}$$

Take the square root of 4.

$$n=\frac{-\left(-8\right)±2}{2\left(-1\right)}$$

The opposite of -8 is 8.

$$n=\frac{8±2}{2\left(-1\right)}$$

Multiply 2 times -1.

$$n=\frac{8±2}{-2}$$

Now solve the equation n=\frac{8±2}{-2} when ± is plus. Add 8 to 2.

$$n=\frac{10}{-2}$$

Divide 10 by -2.

$$n=-5$$

Now solve the equation n=\frac{8±2}{-2} when ± is minus. Subtract 2 from 8.

$$n=\frac{6}{-2}$$

Divide 6 by -2.

$$n=-3$$

The equation is now solved.

$$n=-5 n=-3$$

Steps for Completing the Square

Variable n cannot be equal to any of the values -21,3 since division by zero is not defined. Multiply both sides of the equation by \left(n-3\right)\left(n+21\right), the least common multiple of n-3,n+21.

$$\left(n+21\right)\left(-2\right)=\left(n-3\right)\left(n+9\right)$$

Use the distributive property to multiply n+21 by -2.

$$-2n-42=\left(n-3\right)\left(n+9\right)$$

Use the distributive property to multiply n-3 by n+9 and combine like terms.

$$-2n-42=n^{2}+6n-27$$

Subtract n^{2} from both sides.

$$-2n-42-n^{2}=6n-27$$

Subtract 6n from both sides.

$$-2n-42-n^{2}-6n=-27$$

Combine -2n and -6n to get -8n.

$$-8n-42-n^{2}=-27$$

Add 42 to both sides.

$$-8n-n^{2}=-27+42$$

Add -27 and 42 to get 15.

$$-8n-n^{2}=15$$

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.

$$-n^{2}-8n=15$$

Divide both sides by -1.

$$\frac{-n^{2}-8n}{-1}=\frac{15}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$n^{2}+\left(-\frac{8}{-1}\right)n=\frac{15}{-1}$$

Divide -8 by -1.

$$n^{2}+8n=\frac{15}{-1}$$

Divide 15 by -1.

$$n^{2}+8n=-15$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+8n+4^{2}=-15+4^{2}$$

Square 4.

$$n^{2}+8n+16=-15+16$$

Add -15 to 16.

$$n^{2}+8n+16=1$$

Factor n^{2}+8n+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$n+4=1 n+4=-1$$

Subtract 4 from both sides of the equation.

$$n=-3 n=-5$$