Solve 2-k/k^2+1=5

$$\frac { 2 - k } { k ^ { 2 } + 1 } = 5$$

Solve for k

$$k=\frac{-\sqrt{59}i-1}{10}\approx -0.1-0.768114575i$$

$$k=\frac{-1+\sqrt{59}i}{10}\approx -0.1+0.768114575i$$

Steps Using the Quadratic Formula

Variable k cannot be equal to any of the values -i,i since division by zero is not defined. Multiply both sides of the equation by \left(k-i\right)\left(k+i\right).

$$2-k=5\left(k-i\right)\left(k+i\right)$$

Use the distributive property to multiply 5 by k-i.

$$2-k=\left(5k-5i\right)\left(k+i\right)$$

Use the distributive property to multiply 5k-5i by k+i and combine like terms.

$$2-k=5k^{2}+5$$

Subtract 5k^{2} from both sides.

$$2-k-5k^{2}=5$$

Subtract 5 from both sides.

$$2-k-5k^{2}-5=0$$

Subtract 5 from 2 to get -3.

$$-3-k-5k^{2}=0$$

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.

$$-5k^{2}-k-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, -1 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-\left(-1\right)±\sqrt{1-4\left(-5\right)\left(-3\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$k=\frac{-\left(-1\right)±\sqrt{1+20\left(-3\right)}}{2\left(-5\right)}$$

Multiply 20 times -3.

$$k=\frac{-\left(-1\right)±\sqrt{1-60}}{2\left(-5\right)}$$

Add 1 to -60.

$$k=\frac{-\left(-1\right)±\sqrt{-59}}{2\left(-5\right)}$$

Take the square root of -59.

$$k=\frac{-\left(-1\right)±\sqrt{59}i}{2\left(-5\right)}$$

The opposite of -1 is 1.

$$k=\frac{1±\sqrt{59}i}{2\left(-5\right)}$$

Multiply 2 times -5.

$$k=\frac{1±\sqrt{59}i}{-10}$$

Now solve the equation k=\frac{1±\sqrt{59}i}{-10} when ± is plus. Add 1 to i\sqrt{59}.

$$k=\frac{1+\sqrt{59}i}{-10}$$

Divide 1+i\sqrt{59} by -10.

$$k=\frac{-\sqrt{59}i-1}{10}$$

Now solve the equation k=\frac{1±\sqrt{59}i}{-10} when ± is minus. Subtract i\sqrt{59} from 1.

$$k=\frac{-\sqrt{59}i+1}{-10}$$

Divide 1-i\sqrt{59} by -10.

$$k=\frac{-1+\sqrt{59}i}{10}$$

The equation is now solved.

$$k=\frac{-\sqrt{59}i-1}{10} k=\frac{-1+\sqrt{59}i}{10}$$

Steps for Completing the Square

Variable k cannot be equal to any of the values -i,i since division by zero is not defined. Multiply both sides of the equation by \left(k-i\right)\left(k+i\right).

$$2-k=5\left(k-i\right)\left(k+i\right)$$

Use the distributive property to multiply 5 by k-i.

$$2-k=\left(5k-5i\right)\left(k+i\right)$$

Use the distributive property to multiply 5k-5i by k+i and combine like terms.

$$2-k=5k^{2}+5$$

Subtract 5k^{2} from both sides.

$$2-k-5k^{2}=5$$

Subtract 2 from both sides.

$$-k-5k^{2}=5-2$$

Subtract 2 from 5 to get 3.

$$-k-5k^{2}=3$$

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.

$$-5k^{2}-k=3$$

Divide both sides by -5.

$$\frac{-5k^{2}-k}{-5}=\frac{3}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$k^{2}+\left(-\frac{1}{-5}\right)k=\frac{3}{-5}$$

Divide -1 by -5.

$$k^{2}+\frac{1}{5}k=\frac{3}{-5}$$

Divide 3 by -5.

$$k^{2}+\frac{1}{5}k=-\frac{3}{5}$$

Divide \frac{1}{5}, the coefficient of the x term, by 2 to get \frac{1}{10}. Then add the square of \frac{1}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$k^{2}+\frac{1}{5}k+\left(\frac{1}{10}\right)^{2}=-\frac{3}{5}+\left(\frac{1}{10}\right)^{2}$$

Square \frac{1}{10} by squaring both the numerator and the denominator of the fraction.

$$k^{2}+\frac{1}{5}k+\frac{1}{100}=-\frac{3}{5}+\frac{1}{100}$$

Add -\frac{3}{5} to \frac{1}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$k^{2}+\frac{1}{5}k+\frac{1}{100}=-\frac{59}{100}$$

Factor k^{2}+\frac{1}{5}k+\frac{1}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(k+\frac{1}{10}\right)^{2}=-\frac{59}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k+\frac{1}{10}\right)^{2}}=\sqrt{-\frac{59}{100}}$$

Simplify.

$$k+\frac{1}{10}=\frac{\sqrt{59}i}{10} k+\frac{1}{10}=-\frac{\sqrt{59}i}{10}$$

Subtract \frac{1}{10} from both sides of the equation.

$$k=\frac{-1+\sqrt{59}i}{10} k=\frac{-\sqrt{59}i-1}{10}$$