Solve x^2-2x-3=-x^2+2x+6+8

$$x ^ { 2 } - 2 x - 3 = - x ^ { 2 } + 2 x + 6 + 8$$

Solve for x

$$x=\frac{\sqrt{38}}{2}+1\approx 4.082207001$$

$$x=-\frac{\sqrt{38}}{2}+1\approx -2.082207001$$

Steps Using the Quadratic Formula

Add 6 and 8 to get 14.

$$x^{2}-2x-3=-x^{2}+2x+14$$

Add x^{2} to both sides.

$$x^{2}-2x-3+x^{2}=2x+14$$

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

$$2x^{2}-2x-3=2x+14$$

Subtract 2x from both sides.

$$2x^{2}-2x-3-2x=14$$

Combine -2x and -2x to get -4x.

$$2x^{2}-4x-3=14$$

Subtract 14 from both sides.

$$2x^{2}-4x-3-14=0$$

Subtract 14 from -3 to get -17.

$$2x^{2}-4x-17=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -4 for b, and -17 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\left(-17\right)}}{2\times 2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-17\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\left(-17\right)}}{2\times 2}$$

Multiply -8 times -17.

$$x=\frac{-\left(-4\right)±\sqrt{16+136}}{2\times 2}$$

Add 16 to 136.

$$x=\frac{-\left(-4\right)±\sqrt{152}}{2\times 2}$$

Take the square root of 152.

$$x=\frac{-\left(-4\right)±2\sqrt{38}}{2\times 2}$$

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{38}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{4±2\sqrt{38}}{4}$$

Now solve the equation x=\frac{4±2\sqrt{38}}{4} when ± is plus. Add 4 to 2\sqrt{38}.

$$x=\frac{2\sqrt{38}+4}{4}$$

Divide 4+2\sqrt{38} by 4.

$$x=\frac{\sqrt{38}}{2}+1$$

Now solve the equation x=\frac{4±2\sqrt{38}}{4} when ± is minus. Subtract 2\sqrt{38} from 4.

$$x=\frac{4-2\sqrt{38}}{4}$$

Divide 4-2\sqrt{38} by 4.

$$x=-\frac{\sqrt{38}}{2}+1$$

The equation is now solved.

$$x=\frac{\sqrt{38}}{2}+1 x=-\frac{\sqrt{38}}{2}+1$$

Steps for Completing the Square

Add 6 and 8 to get 14.

$$x^{2}-2x-3=-x^{2}+2x+14$$

Add x^{2} to both sides.

$$x^{2}-2x-3+x^{2}=2x+14$$

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

$$2x^{2}-2x-3=2x+14$$

Subtract 2x from both sides.

$$2x^{2}-2x-3-2x=14$$

Combine -2x and -2x to get -4x.

$$2x^{2}-4x-3=14$$

Add 3 to both sides.

$$2x^{2}-4x=14+3$$

Add 14 and 3 to get 17.

$$2x^{2}-4x=17$$

Divide both sides by 2.

$$\frac{2x^{2}-4x}{2}=\frac{17}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{4}{2}\right)x=\frac{17}{2}$$

Divide -4 by 2.

$$x^{2}-2x=\frac{17}{2}$$

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.

$$x^{2}-2x+1=\frac{17}{2}+1$$

Add \frac{17}{2} to 1.

$$x^{2}-2x+1=\frac{19}{2}$$

Factor x^{2}-2x+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}.

$$\left(x-1\right)^{2}=\frac{19}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{19}{2}}$$

Simplify.

$$x-1=\frac{\sqrt{38}}{2} x-1=-\frac{\sqrt{38}}{2}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{38}}{2}+1 x=-\frac{\sqrt{38}}{2}+1$$