Solve (x-2)-x+2/3-x-2/2=(x-2)^2-4

$$( x - 2 ) - \frac { x + 2 } { 3 } - \frac { x - 2 } { 2 } = ( x - 2 ) ^ { 2 } - 4$$

Solve for x

$$x = \frac{\sqrt{385} + 25}{12} \approx 3.718451406$$

$$x=\frac{25-\sqrt{385}}{12}\approx 0.448215261$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 6, the least common multiple of 3,2.

$$6x-12-2\left(x+2\right)-3\left(x-2\right)=6\left(x-2\right)^{2}-24$$

Use the distributive property to multiply -2 by x+2.

$$6x-12-2x-4-3\left(x-2\right)=6\left(x-2\right)^{2}-24$$

Combine 6x and -2x to get 4x.

$$4x-12-4-3\left(x-2\right)=6\left(x-2\right)^{2}-24$$

Subtract 4 from -12 to get -16.

$$4x-16-3\left(x-2\right)=6\left(x-2\right)^{2}-24$$

Use the distributive property to multiply -3 by x-2.

$$4x-16-3x+6=6\left(x-2\right)^{2}-24$$

Combine 4x and -3x to get x.

$$x-16+6=6\left(x-2\right)^{2}-24$$

Add -16 and 6 to get -10.

$$x-10=6\left(x-2\right)^{2}-24$$

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

$$x-10=6\left(x^{2}-4x+4\right)-24$$

Use the distributive property to multiply 6 by x^{2}-4x+4.

$$x-10=6x^{2}-24x+24-24$$

Subtract 24 from 24 to get 0.

$$x-10=6x^{2}-24x$$

Subtract 6x^{2} from both sides.

$$x-10-6x^{2}=-24x$$

Add 24x to both sides.

$$x-10-6x^{2}+24x=0$$

Combine x and 24x to get 25x.

$$25x-10-6x^{2}=0$$

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.

$$-6x^{2}+25x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 25 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-25±\sqrt{25^{2}-4\left(-6\right)\left(-10\right)}}{2\left(-6\right)}$$

Square 25.

$$x=\frac{-25±\sqrt{625-4\left(-6\right)\left(-10\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-25±\sqrt{625+24\left(-10\right)}}{2\left(-6\right)}$$

Multiply 24 times -10.

$$x=\frac{-25±\sqrt{625-240}}{2\left(-6\right)}$$

Add 625 to -240.

$$x=\frac{-25±\sqrt{385}}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-25±\sqrt{385}}{-12}$$

Now solve the equation x=\frac{-25±\sqrt{385}}{-12} when ± is plus. Add -25 to \sqrt{385}.

$$x=\frac{\sqrt{385}-25}{-12}$$

Divide -25+\sqrt{385} by -12.

$$x=\frac{25-\sqrt{385}}{12}$$

Now solve the equation x=\frac{-25±\sqrt{385}}{-12} when ± is minus. Subtract \sqrt{385} from -25.

$$x=\frac{-\sqrt{385}-25}{-12}$$

Divide -25-\sqrt{385} by -12.

$$x=\frac{\sqrt{385}+25}{12}$$

The equation is now solved.

$$x=\frac{25-\sqrt{385}}{12} x=\frac{\sqrt{385}+25}{12}$$

Steps for Completing the Square

Multiply both sides of the equation by 6, the least common multiple of 3,2.

$$6x-12-2\left(x+2\right)-3\left(x-2\right)=6\left(x-2\right)^{2}-24$$

Use the distributive property to multiply -2 by x+2.

$$6x-12-2x-4-3\left(x-2\right)=6\left(x-2\right)^{2}-24$$

Combine 6x and -2x to get 4x.

$$4x-12-4-3\left(x-2\right)=6\left(x-2\right)^{2}-24$$

Subtract 4 from -12 to get -16.

$$4x-16-3\left(x-2\right)=6\left(x-2\right)^{2}-24$$

Use the distributive property to multiply -3 by x-2.

$$4x-16-3x+6=6\left(x-2\right)^{2}-24$$

Combine 4x and -3x to get x.

$$x-16+6=6\left(x-2\right)^{2}-24$$

Add -16 and 6 to get -10.

$$x-10=6\left(x-2\right)^{2}-24$$

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

$$x-10=6\left(x^{2}-4x+4\right)-24$$

Use the distributive property to multiply 6 by x^{2}-4x+4.

$$x-10=6x^{2}-24x+24-24$$

Subtract 24 from 24 to get 0.

$$x-10=6x^{2}-24x$$

Subtract 6x^{2} from both sides.

$$x-10-6x^{2}=-24x$$

Add 24x to both sides.

$$x-10-6x^{2}+24x=0$$

Combine x and 24x to get 25x.

$$25x-10-6x^{2}=0$$

Add 10 to both sides. Anything plus zero gives itself.

$$25x-6x^{2}=10$$

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.

$$-6x^{2}+25x=10$$

Divide both sides by -6.

$$\frac{-6x^{2}+25x}{-6}=\frac{10}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{25}{-6}x=\frac{10}{-6}$$

Divide 25 by -6.

$$x^{2}-\frac{25}{6}x=\frac{10}{-6}$$

Reduce the fraction \frac{10}{-6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{25}{6}x=-\frac{5}{3}$$

Divide -\frac{25}{6}, the coefficient of the x term, by 2 to get -\frac{25}{12}. Then add the square of -\frac{25}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{25}{6}x+\left(-\frac{25}{12}\right)^{2}=-\frac{5}{3}+\left(-\frac{25}{12}\right)^{2}$$

Square -\frac{25}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{25}{6}x+\frac{625}{144}=-\frac{5}{3}+\frac{625}{144}$$

Add -\frac{5}{3} to \frac{625}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{25}{6}x+\frac{625}{144}=\frac{385}{144}$$

Factor x^{2}-\frac{25}{6}x+\frac{625}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25}{12}\right)^{2}=\frac{385}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{12}\right)^{2}}=\sqrt{\frac{385}{144}}$$

Simplify.

$$x-\frac{25}{12}=\frac{\sqrt{385}}{12} x-\frac{25}{12}=-\frac{\sqrt{385}}{12}$$

Add \frac{25}{12} to both sides of the equation.

$$x=\frac{\sqrt{385}+25}{12} x=\frac{25-\sqrt{385}}{12}$$