Solve (-2-k)^2+k^2-4k+4+1=20

$$( - 2 - k ) ^ { 2 } + k ^ { 2 } - 4 k + 4 + 1 = 20$$

Solve for k

$$k = \frac{\sqrt{22}}{2} \approx 2.34520788$$

$$k = -\frac{\sqrt{22}}{2} \approx -2.34520788$$

Steps by Finding Square Root

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-2-k\right)^{2}.

$$4+4k+k^{2}+k^{2}-4k+4+1=20$$

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

$$4+4k+2k^{2}-4k+4+1=20$$

Combine 4k and -4k to get 0.

$$4+2k^{2}+4+1=20$$

Add 4 and 4 to get 8.

$$8+2k^{2}+1=20$$

Add 8 and 1 to get 9.

$$9+2k^{2}=20$$

Subtract 9 from both sides.

$$2k^{2}=20-9$$

Subtract 9 from 20 to get 11.

$$2k^{2}=11$$

Divide both sides by 2.

$$k^{2}=\frac{11}{2}$$

Take the square root of both sides of the equation.

$$k=\frac{\sqrt{22}}{2} k=-\frac{\sqrt{22}}{2}$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-2-k\right)^{2}.

$$4+4k+k^{2}+k^{2}-4k+4+1=20$$

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

$$4+4k+2k^{2}-4k+4+1=20$$

Combine 4k and -4k to get 0.

$$4+2k^{2}+4+1=20$$

Add 4 and 4 to get 8.

$$8+2k^{2}+1=20$$

Add 8 and 1 to get 9.

$$9+2k^{2}=20$$

Subtract 20 from both sides.

$$9+2k^{2}-20=0$$

Subtract 20 from 9 to get -11.

$$-11+2k^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$2k^{2}-11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 0 for b, and -11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{0±\sqrt{0^{2}-4\times 2\left(-11\right)}}{2\times 2}$$

Square 0.

$$k=\frac{0±\sqrt{-4\times 2\left(-11\right)}}{2\times 2}$$

Multiply -4 times 2.

$$k=\frac{0±\sqrt{-8\left(-11\right)}}{2\times 2}$$

Multiply -8 times -11.

$$k=\frac{0±\sqrt{88}}{2\times 2}$$

Take the square root of 88.

$$k=\frac{0±2\sqrt{22}}{2\times 2}$$

Multiply 2 times 2.

$$k=\frac{0±2\sqrt{22}}{4}$$

Now solve the equation k=\frac{0±2\sqrt{22}}{4} when ± is plus.

$$k=\frac{\sqrt{22}}{2}$$

Now solve the equation k=\frac{0±2\sqrt{22}}{4} when ± is minus.

$$k=-\frac{\sqrt{22}}{2}$$

The equation is now solved.

$$k=\frac{\sqrt{22}}{2} k=-\frac{\sqrt{22}}{2}$$