Solve n-4/n+6/n+1=5/4

$$\frac { n - 4 } { n } + \frac { 6 } { n + 1 } = \frac { 5 } { 4 }$$

Solve for n

$$n=\frac{-\sqrt{15}i+7}{2}\approx 3.5-1.936491673i$$

$$n=\frac{7+\sqrt{15}i}{2}\approx 3.5+1.936491673i$$

Steps Using the Quadratic Formula

Variable n cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by 4n\left(n+1\right), the least common multiple of n,n+1,4.

$$\left(4n+4\right)\left(n-4\right)+4n\times 6=5n\left(n+1\right)$$

Use the distributive property to multiply 4n+4 by n-4 and combine like terms.

$$4n^{2}-12n-16+4n\times 6=5n\left(n+1\right)$$

Multiply 4 and 6 to get 24.

$$4n^{2}-12n-16+24n=5n\left(n+1\right)$$

Combine -12n and 24n to get 12n.

$$4n^{2}+12n-16=5n\left(n+1\right)$$

Use the distributive property to multiply 5n by n+1.

$$4n^{2}+12n-16=5n^{2}+5n$$

Subtract 5n^{2} from both sides.

$$4n^{2}+12n-16-5n^{2}=5n$$

Combine 4n^{2} and -5n^{2} to get -n^{2}.

$$-n^{2}+12n-16=5n$$

Subtract 5n from both sides.

$$-n^{2}+12n-16-5n=0$$

Combine 12n and -5n to get 7n.

$$-n^{2}+7n-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 7 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-7±\sqrt{7^{2}-4\left(-1\right)\left(-16\right)}}{2\left(-1\right)}$$

Square 7.

$$n=\frac{-7±\sqrt{49-4\left(-1\right)\left(-16\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$n=\frac{-7±\sqrt{49+4\left(-16\right)}}{2\left(-1\right)}$$

Multiply 4 times -16.

$$n=\frac{-7±\sqrt{49-64}}{2\left(-1\right)}$$

Add 49 to -64.

$$n=\frac{-7±\sqrt{-15}}{2\left(-1\right)}$$

Take the square root of -15.

$$n=\frac{-7±\sqrt{15}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$n=\frac{-7±\sqrt{15}i}{-2}$$

Now solve the equation n=\frac{-7±\sqrt{15}i}{-2} when ± is plus. Add -7 to i\sqrt{15}.

$$n=\frac{-7+\sqrt{15}i}{-2}$$

Divide -7+i\sqrt{15} by -2.

$$n=\frac{-\sqrt{15}i+7}{2}$$

Now solve the equation n=\frac{-7±\sqrt{15}i}{-2} when ± is minus. Subtract i\sqrt{15} from -7.

$$n=\frac{-\sqrt{15}i-7}{-2}$$

Divide -7-i\sqrt{15} by -2.

$$n=\frac{7+\sqrt{15}i}{2}$$

The equation is now solved.

$$n=\frac{-\sqrt{15}i+7}{2} n=\frac{7+\sqrt{15}i}{2}$$

Steps for Completing the Square

Variable n cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by 4n\left(n+1\right), the least common multiple of n,n+1,4.

$$\left(4n+4\right)\left(n-4\right)+4n\times 6=5n\left(n+1\right)$$

Use the distributive property to multiply 4n+4 by n-4 and combine like terms.

$$4n^{2}-12n-16+4n\times 6=5n\left(n+1\right)$$

Multiply 4 and 6 to get 24.

$$4n^{2}-12n-16+24n=5n\left(n+1\right)$$

Combine -12n and 24n to get 12n.

$$4n^{2}+12n-16=5n\left(n+1\right)$$

Use the distributive property to multiply 5n by n+1.

$$4n^{2}+12n-16=5n^{2}+5n$$

Subtract 5n^{2} from both sides.

$$4n^{2}+12n-16-5n^{2}=5n$$

Combine 4n^{2} and -5n^{2} to get -n^{2}.

$$-n^{2}+12n-16=5n$$

Subtract 5n from both sides.

$$-n^{2}+12n-16-5n=0$$

Combine 12n and -5n to get 7n.

$$-n^{2}+7n-16=0$$

Add 16 to both sides. Anything plus zero gives itself.

$$-n^{2}+7n=16$$

Divide both sides by -1.

$$\frac{-n^{2}+7n}{-1}=\frac{16}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$n^{2}+\frac{7}{-1}n=\frac{16}{-1}$$

Divide 7 by -1.

$$n^{2}-7n=\frac{16}{-1}$$

Divide 16 by -1.

$$n^{2}-7n=-16$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-7n+\left(-\frac{7}{2}\right)^{2}=-16+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-7n+\frac{49}{4}=-16+\frac{49}{4}$$

Add -16 to \frac{49}{4}.

$$n^{2}-7n+\frac{49}{4}=-\frac{15}{4}$$

Factor n^{2}-7n+\frac{49}{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}.

$$\left(n-\frac{7}{2}\right)^{2}=-\frac{15}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{7}{2}\right)^{2}}=\sqrt{-\frac{15}{4}}$$

Simplify.

$$n-\frac{7}{2}=\frac{\sqrt{15}i}{2} n-\frac{7}{2}=-\frac{\sqrt{15}i}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$n=\frac{7+\sqrt{15}i}{2} n=\frac{-\sqrt{15}i+7}{2}$$