Solve 2x+1/3-4x-x^2/12=x^2-4/9

$$\frac { 2 x + 1 } { 3 } - \frac { 4 x - x ^ { 2 } } { 12 } = \frac { x ^ { 2 } - 4 } { 9 }$$

Solve for x

$$x=-2$$

$$x=14$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 36, the least common multiple of 3,12,9.

$$12\left(2x+1\right)-3\left(4x-x^{2}\right)=4\left(x^{2}-4\right)$$

Use the distributive property to multiply 12 by 2x+1.

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

Use the distributive property to multiply -3 by 4x-x^{2}.

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

Combine 24x and -12x to get 12x.

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

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

$$12x+12+3x^{2}=4x^{2}-16$$

Subtract 4x^{2} from both sides.

$$12x+12+3x^{2}-4x^{2}=-16$$

Combine 3x^{2} and -4x^{2} to get -x^{2}.

$$12x+12-x^{2}=-16$$

Add 16 to both sides.

$$12x+12-x^{2}+16=0$$

Add 12 and 16 to get 28.

$$12x+28-x^{2}=0$$

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

$$-x^{2}+12x+28=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+28. To find a and b, set up a system to be solved.

$$a+b=12 ab=-28=-28$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -28.

$$-1,28 -2,14 -4,7$$

Calculate the sum for each pair.

$$-1+28=27 -2+14=12 -4+7=3$$

The solution is the pair that gives sum 12.

$$a=14 b=-2$$

Rewrite -x^{2}+12x+28 as \left(-x^{2}+14x\right)+\left(-2x+28\right).

$$\left(-x^{2}+14x\right)+\left(-2x+28\right)$$

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

$$-x\left(x-14\right)-2\left(x-14\right)$$

Factor out common term x-14 by using distributive property.

$$\left(x-14\right)\left(-x-2\right)$$

To find equation solutions, solve x-14=0 and -x-2=0.

$$x=14 x=-2$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 36, the least common multiple of 3,12,9.

$$12\left(2x+1\right)-3\left(4x-x^{2}\right)=4\left(x^{2}-4\right)$$

Use the distributive property to multiply 12 by 2x+1.

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

Use the distributive property to multiply -3 by 4x-x^{2}.

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

Combine 24x and -12x to get 12x.

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

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

$$12x+12+3x^{2}=4x^{2}-16$$

Subtract 4x^{2} from both sides.

$$12x+12+3x^{2}-4x^{2}=-16$$

Combine 3x^{2} and -4x^{2} to get -x^{2}.

$$12x+12-x^{2}=-16$$

Add 16 to both sides.

$$12x+12-x^{2}+16=0$$

Add 12 and 16 to get 28.

$$12x+28-x^{2}=0$$

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.

$$-x^{2}+12x+28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 12 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-1\right)\times 28}}{2\left(-1\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-1\right)\times 28}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-12±\sqrt{144+4\times 28}}{2\left(-1\right)}$$

Multiply 4 times 28.

$$x=\frac{-12±\sqrt{144+112}}{2\left(-1\right)}$$

Add 144 to 112.

$$x=\frac{-12±\sqrt{256}}{2\left(-1\right)}$$

Take the square root of 256.

$$x=\frac{-12±16}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-12±16}{-2}$$

Now solve the equation x=\frac{-12±16}{-2} when ± is plus. Add -12 to 16.

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

Divide 4 by -2.

$$x=-2$$

Now solve the equation x=\frac{-12±16}{-2} when ± is minus. Subtract 16 from -12.

$$x=-\frac{28}{-2}$$

Divide -28 by -2.

$$x=14$$

The equation is now solved.

$$x=-2 x=14$$

Steps for Completing the Square

Multiply both sides of the equation by 36, the least common multiple of 3,12,9.

$$12\left(2x+1\right)-3\left(4x-x^{2}\right)=4\left(x^{2}-4\right)$$

Use the distributive property to multiply 12 by 2x+1.

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

Use the distributive property to multiply -3 by 4x-x^{2}.

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

Combine 24x and -12x to get 12x.

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

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

$$12x+12+3x^{2}=4x^{2}-16$$

Subtract 4x^{2} from both sides.

$$12x+12+3x^{2}-4x^{2}=-16$$

Combine 3x^{2} and -4x^{2} to get -x^{2}.

$$12x+12-x^{2}=-16$$

Subtract 12 from both sides.

$$12x-x^{2}=-16-12$$

Subtract 12 from -16 to get -28.

$$12x-x^{2}=-28$$

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.

$$-x^{2}+12x=-28$$

Divide both sides by -1.

$$\frac{-x^{2}+12x}{-1}=-\frac{28}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{12}{-1}x=-\frac{28}{-1}$$

Divide 12 by -1.

$$x^{2}-12x=-\frac{28}{-1}$$

Divide -28 by -1.

$$x^{2}-12x=28$$

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

$$x^{2}-12x+\left(-6\right)^{2}=28+\left(-6\right)^{2}$$

Square -6.

$$x^{2}-12x+36=28+36$$

Add 28 to 36.

$$x^{2}-12x+36=64$$

Factor x^{2}-12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-6\right)^{2}=64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{64}$$

Simplify.

$$x-6=8 x-6=-8$$

Add 6 to both sides of the equation.

$$x=14 x=-2$$