Solve 8/k=2+k

$$\frac { 8 } { k } = 2 + k$$

Solve for k

$$k=-4$$

$$k=2$$

Steps Using the Quadratic Formula

Variable k cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by k.

$$8=k\times 2+kk$$

Multiply k and k to get k^{2}.

$$8=k\times 2+k^{2}$$

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

$$k\times 2+k^{2}=8$$

Subtract 8 from both sides.

$$k\times 2+k^{2}-8=0$$

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.

$$k^{2}+2k-8=0$$

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

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

Square 2.

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

Multiply -4 times -8.

$$k=\frac{-2±\sqrt{4+32}}{2}$$

Add 4 to 32.

$$k=\frac{-2±\sqrt{36}}{2}$$

Take the square root of 36.

$$k=\frac{-2±6}{2}$$

Now solve the equation k=\frac{-2±6}{2} when ± is plus. Add -2 to 6.

$$k=\frac{4}{2}$$

Divide 4 by 2.

$$k=2$$

Now solve the equation k=\frac{-2±6}{2} when ± is minus. Subtract 6 from -2.

$$k=-\frac{8}{2}$$

Divide -8 by 2.

$$k=-4$$

The equation is now solved.

$$k=2 k=-4$$

Steps for Completing the Square

Variable k cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by k.

$$8=k\times 2+kk$$

Multiply k and k to get k^{2}.

$$8=k\times 2+k^{2}$$

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

$$k\times 2+k^{2}=8$$

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.

$$k^{2}+2k=8$$

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

$$k^{2}+2k+1^{2}=8+1^{2}$$

Square 1.

$$k^{2}+2k+1=8+1$$

Add 8 to 1.

$$k^{2}+2k+1=9$$

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

$$\left(k+1\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k+1\right)^{2}}=\sqrt{9}$$

Simplify.

$$k+1=3 k+1=-3$$

Subtract 1 from both sides of the equation.

$$k=2 k=-4$$