Solve (1-3)/3+2x-5x+2/2-(x-2)(x+2)/3=5/6x+frac{2}{3}

$$\frac { ( 1 - 3 ) } { 3 } + 2 x - \frac { 5 x + 2 } { 2 } - \frac { ( x - 2 ) ( x + 2 ) } { 3 } = \frac { 5 } { 6 } x + \frac { 2 } { 3 }$$

Solve for x

$$x=-3$$

$$x=-1$$

Steps Using the Quadratic Formula

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

$$2\left(1-3\right)+12x-3\left(5x+2\right)-2\left(x-2\right)\left(x+2\right)=5x+4$$

Subtract 3 from 1 to get -2.

$$2\left(-2\right)+12x-3\left(5x+2\right)-2\left(x-2\right)\left(x+2\right)=5x+4$$

Multiply 2 and -2 to get -4.

$$-4+12x-3\left(5x+2\right)-2\left(x-2\right)\left(x+2\right)=5x+4$$

Use the distributive property to multiply -3 by 5x+2.

$$-4+12x-15x-6-2\left(x-2\right)\left(x+2\right)=5x+4$$

Combine 12x and -15x to get -3x.

$$-4-3x-6-2\left(x-2\right)\left(x+2\right)=5x+4$$

Subtract 6 from -4 to get -10.

$$-10-3x-2\left(x-2\right)\left(x+2\right)=5x+4$$

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

$$-10-3x+\left(-2x+4\right)\left(x+2\right)=5x+4$$

Use the distributive property to multiply -2x+4 by x+2 and combine like terms.

$$-10-3x-2x^{2}+8=5x+4$$

Add -10 and 8 to get -2.

$$-2-3x-2x^{2}=5x+4$$

Subtract 5x from both sides.

$$-2-3x-2x^{2}-5x=4$$

Combine -3x and -5x to get -8x.

$$-2-8x-2x^{2}=4$$

Subtract 4 from both sides.

$$-2-8x-2x^{2}-4=0$$

Subtract 4 from -2 to get -6.

$$-6-8x-2x^{2}=0$$

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.

$$-2x^{2}-8x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -8 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-2\right)\left(-6\right)}}{2\left(-2\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-2\right)\left(-6\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-8\right)±\sqrt{64+8\left(-6\right)}}{2\left(-2\right)}$$

Multiply 8 times -6.

$$x=\frac{-\left(-8\right)±\sqrt{64-48}}{2\left(-2\right)}$$

Add 64 to -48.

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

Take the square root of 16.

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

The opposite of -8 is 8.

$$x=\frac{8±4}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{8±4}{-4}$$

Now solve the equation x=\frac{8±4}{-4} when ± is plus. Add 8 to 4.

$$x=\frac{12}{-4}$$

Divide 12 by -4.

$$x=-3$$

Now solve the equation x=\frac{8±4}{-4} when ± is minus. Subtract 4 from 8.

$$x=\frac{4}{-4}$$

Divide 4 by -4.

$$x=-1$$

The equation is now solved.

$$x=-3 x=-1$$

Steps for Completing the Square

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

$$2\left(1-3\right)+12x-3\left(5x+2\right)-2\left(x-2\right)\left(x+2\right)=5x+4$$

Subtract 3 from 1 to get -2.

$$2\left(-2\right)+12x-3\left(5x+2\right)-2\left(x-2\right)\left(x+2\right)=5x+4$$

Multiply 2 and -2 to get -4.

$$-4+12x-3\left(5x+2\right)-2\left(x-2\right)\left(x+2\right)=5x+4$$

Use the distributive property to multiply -3 by 5x+2.

$$-4+12x-15x-6-2\left(x-2\right)\left(x+2\right)=5x+4$$

Combine 12x and -15x to get -3x.

$$-4-3x-6-2\left(x-2\right)\left(x+2\right)=5x+4$$

Subtract 6 from -4 to get -10.

$$-10-3x-2\left(x-2\right)\left(x+2\right)=5x+4$$

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

$$-10-3x+\left(-2x+4\right)\left(x+2\right)=5x+4$$

Use the distributive property to multiply -2x+4 by x+2 and combine like terms.

$$-10-3x-2x^{2}+8=5x+4$$

Add -10 and 8 to get -2.

$$-2-3x-2x^{2}=5x+4$$

Subtract 5x from both sides.

$$-2-3x-2x^{2}-5x=4$$

Combine -3x and -5x to get -8x.

$$-2-8x-2x^{2}=4$$

Add 2 to both sides.

$$-8x-2x^{2}=4+2$$

Add 4 and 2 to get 6.

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

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.

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

Divide both sides by -2.

$$\frac{-2x^{2}-8x}{-2}=\frac{6}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{8}{-2}\right)x=\frac{6}{-2}$$

Divide -8 by -2.

$$x^{2}+4x=\frac{6}{-2}$$

Divide 6 by -2.

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

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square 2.

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

Add -3 to 4.

$$x^{2}+4x+4=1$$

Factor x^{2}+4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+2=1 x+2=-1$$

Subtract 2 from both sides of the equation.

$$x=-1 x=-3$$