Solve 2(x+6)/4-x^2-2/x+2=3

$$\frac { 2 ( x + 6 ) } { 4 - x ^ { 2 } } - \frac { 2 } { x + 2 } = 3$$

Solve for x

$$x=\frac{2}{3}\approx 0.666666667$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of 4-x^{2},x+2.

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

Use the distributive property to multiply -2 by x+6.

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

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

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

To find the opposite of 2x-4, find the opposite of each term.

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

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

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

Add -12 and 4 to get -8.

$$-4x-8=3\left(x-2\right)\left(x+2\right)$$

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

$$-4x-8=\left(3x-6\right)\left(x+2\right)$$

Use the distributive property to multiply 3x-6 by x+2 and combine like terms.

$$-4x-8=3x^{2}-12$$

Subtract 3x^{2} from both sides.

$$-4x-8-3x^{2}=-12$$

Add 12 to both sides.

$$-4x-8-3x^{2}+12=0$$

Add -8 and 12 to get 4.

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

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -3x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=-4 ab=-3\times 4=-12$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -12.

$$1,-12 2,-6 3,-4$$

Calculate the sum for each pair.

$$1-12=-11 2-6=-4 3-4=-1$$

The solution is the pair that gives sum -4.

$$a=2 b=-6$$

Rewrite -3x^{2}-4x+4 as \left(-3x^{2}+2x\right)+\left(-6x+4\right).

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

Factor out -x in the first and -2 in the second group.

$$-x\left(3x-2\right)-2\left(3x-2\right)$$

Factor out common term 3x-2 by using distributive property.

$$\left(3x-2\right)\left(-x-2\right)$$

To find equation solutions, solve 3x-2=0 and -x-2=0.

$$x=\frac{2}{3} x=-2$$

Variable x cannot be equal to -2.

$$x=\frac{2}{3}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of 4-x^{2},x+2.

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

Use the distributive property to multiply -2 by x+6.

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

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

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

To find the opposite of 2x-4, find the opposite of each term.

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

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

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

Add -12 and 4 to get -8.

$$-4x-8=3\left(x-2\right)\left(x+2\right)$$

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

$$-4x-8=\left(3x-6\right)\left(x+2\right)$$

Use the distributive property to multiply 3x-6 by x+2 and combine like terms.

$$-4x-8=3x^{2}-12$$

Subtract 3x^{2} from both sides.

$$-4x-8-3x^{2}=-12$$

Add 12 to both sides.

$$-4x-8-3x^{2}+12=0$$

Add -8 and 12 to get 4.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -4 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-3\right)\times 4}}{2\left(-3\right)}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-3\right)\times 4}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-4\right)±\sqrt{16+12\times 4}}{2\left(-3\right)}$$

Multiply 12 times 4.

$$x=\frac{-\left(-4\right)±\sqrt{16+48}}{2\left(-3\right)}$$

Add 16 to 48.

$$x=\frac{-\left(-4\right)±\sqrt{64}}{2\left(-3\right)}$$

Take the square root of 64.

$$x=\frac{-\left(-4\right)±8}{2\left(-3\right)}$$

The opposite of -4 is 4.

$$x=\frac{4±8}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{4±8}{-6}$$

Now solve the equation x=\frac{4±8}{-6} when ± is plus. Add 4 to 8.

$$x=\frac{12}{-6}$$

Divide 12 by -6.

$$x=-2$$

Now solve the equation x=\frac{4±8}{-6} when ± is minus. Subtract 8 from 4.

$$x=-\frac{4}{-6}$$

Reduce the fraction \frac{-4}{-6} to lowest terms by extracting and canceling out 2.

$$x=\frac{2}{3}$$

The equation is now solved.

$$x=-2 x=\frac{2}{3}$$

Variable x cannot be equal to -2.

$$x=\frac{2}{3}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of 4-x^{2},x+2.

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

Use the distributive property to multiply -2 by x+6.

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

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

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

To find the opposite of 2x-4, find the opposite of each term.

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

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

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

Add -12 and 4 to get -8.

$$-4x-8=3\left(x-2\right)\left(x+2\right)$$

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

$$-4x-8=\left(3x-6\right)\left(x+2\right)$$

Use the distributive property to multiply 3x-6 by x+2 and combine like terms.

$$-4x-8=3x^{2}-12$$

Subtract 3x^{2} from both sides.

$$-4x-8-3x^{2}=-12$$

Add 8 to both sides.

$$-4x-3x^{2}=-12+8$$

Add -12 and 8 to get -4.

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

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.

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

Divide both sides by -3.

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

Dividing by -3 undoes the multiplication by -3.

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

Divide -4 by -3.

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

Divide -4 by -3.

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

Divide \frac{4}{3}, the coefficient of the x term, by 2 to get \frac{2}{3}. Then add the square of \frac{2}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{2}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=\frac{4}{3}+\frac{4}{9}$$

Add \frac{4}{3} to \frac{4}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=\frac{16}{9}$$

Factor x^{2}+\frac{4}{3}x+\frac{4}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{2}{3}\right)^{2}=\frac{16}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{2}{3}\right)^{2}}=\sqrt{\frac{16}{9}}$$

Simplify.

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

Subtract \frac{2}{3} from both sides of the equation.

$$x=\frac{2}{3} x=-2$$

Variable x cannot be equal to -2.

$$x=\frac{2}{3}$$