Solve -2k-1=-k^2-1

$$- 2 k - 1 = - k ^ { 2 } - 1$$

Solve for k

$$k=2$$

$$k=0$$

Steps Using Factoring

Add k^{2} to both sides.

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

Add 1 to both sides.

$$-2k-1+k^{2}+1=0$$

Add -1 and 1 to get 0.

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

Factor out k.

$$k\left(-2+k\right)=0$$

To find equation solutions, solve k=0 and -2+k=0.

$$k=0 k=2$$

Steps Using the Quadratic Formula

Add k^{2} to both sides.

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

Add 1 to both sides.

$$-2k-1+k^{2}+1=0$$

Add -1 and 1 to get 0.

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

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.

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

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

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

Take the square root of \left(-2\right)^{2}.

$$k=\frac{-\left(-2\right)±2}{2}$$

The opposite of -2 is 2.

$$k=\frac{2±2}{2}$$

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

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

Divide 4 by 2.

$$k=2$$

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

$$k=\frac{0}{2}$$

Divide 0 by 2.

$$k=0$$

The equation is now solved.

$$k=2 k=0$$

Steps for Completing the Square

Add k^{2} to both sides.

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

Add 1 to both sides.

$$-2k-1+k^{2}+1=0$$

Add -1 and 1 to get 0.

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

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.

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

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.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$k-1=1 k-1=-1$$

Add 1 to both sides of the equation.

$$k=2 k=0$$