Solve 2x-(x+1)^2=5

$$2 x - ( x + 1 ) ^ { 2 } = 5$$

Solve for x (complex solution)

$$x=-\sqrt{6}i\approx -0-2.449489743i$$

$$x=\sqrt{6}i\approx 2.449489743i$$

Steps by Finding Square Root

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$2x-\left(x^{2}+2x+1\right)=5$$

To find the opposite of x^{2}+2x+1, find the opposite of each term.

$$2x-x^{2}-2x-1=5$$

Combine 2x and -2x to get 0.

$$-x^{2}-1=5$$

Add 1 to both sides.

$$-x^{2}=5+1$$

Add 5 and 1 to get 6.

$$-x^{2}=6$$

Divide both sides by -1.

$$x^{2}=-6$$

The equation is now solved.

$$x=\sqrt{6}i x=-\sqrt{6}i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$2x-\left(x^{2}+2x+1\right)=5$$

To find the opposite of x^{2}+2x+1, find the opposite of each term.

$$2x-x^{2}-2x-1=5$$

Combine 2x and -2x to get 0.

$$-x^{2}-1=5$$

Subtract 5 from both sides.

$$-x^{2}-1-5=0$$

Subtract 5 from -1 to get -6.

$$-x^{2}-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 0 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{0±\sqrt{4\left(-6\right)}}{2\left(-1\right)}$$

Multiply 4 times -6.

$$x=\frac{0±\sqrt{-24}}{2\left(-1\right)}$$

Take the square root of -24.

$$x=\frac{0±2\sqrt{6}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{0±2\sqrt{6}i}{-2}$$

Now solve the equation x=\frac{0±2\sqrt{6}i}{-2} when ± is plus.

$$x=-\sqrt{6}i$$

Now solve the equation x=\frac{0±2\sqrt{6}i}{-2} when ± is minus.

$$x=\sqrt{6}i$$

The equation is now solved.

$$x=-\sqrt{6}i x=\sqrt{6}i$$