Solve 2+x+4/3=4x+4/3+2-x/x-3

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

Solve for x

$$x=2$$

$$x=4$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 3 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-3\right), the least common multiple of 3,x-3.

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

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

$$\left(3x-9\right)\times 2+\left(x-3\right)\left(x+4\right)=\left(x-3\right)\left(4x+4\right)+3\left(2-x\right)$$

Use the distributive property to multiply 3x-9 by 2.

$$6x-18+\left(x-3\right)\left(x+4\right)=\left(x-3\right)\left(4x+4\right)+3\left(2-x\right)$$

Use the distributive property to multiply x-3 by x+4 and combine like terms.

$$6x-18+x^{2}+x-12=\left(x-3\right)\left(4x+4\right)+3\left(2-x\right)$$

Combine 6x and x to get 7x.

$$7x-18+x^{2}-12=\left(x-3\right)\left(4x+4\right)+3\left(2-x\right)$$

Subtract 12 from -18 to get -30.

$$7x-30+x^{2}=\left(x-3\right)\left(4x+4\right)+3\left(2-x\right)$$

Use the distributive property to multiply x-3 by 4x+4 and combine like terms.

$$7x-30+x^{2}=4x^{2}-8x-12+3\left(2-x\right)$$

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

$$7x-30+x^{2}=4x^{2}-8x-12+6-3x$$

Add -12 and 6 to get -6.

$$7x-30+x^{2}=4x^{2}-8x-6-3x$$

Combine -8x and -3x to get -11x.

$$7x-30+x^{2}=4x^{2}-11x-6$$

Subtract 4x^{2} from both sides.

$$7x-30+x^{2}-4x^{2}=-11x-6$$

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

$$7x-30-3x^{2}=-11x-6$$

Add 11x to both sides.

$$7x-30-3x^{2}+11x=-6$$

Combine 7x and 11x to get 18x.

$$18x-30-3x^{2}=-6$$

Add 6 to both sides.

$$18x-30-3x^{2}+6=0$$

Add -30 and 6 to get -24.

$$18x-24-3x^{2}=0$$

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.

$$-3x^{2}+18x-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 18 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-3\right)\left(-24\right)}}{2\left(-3\right)}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\left(-3\right)\left(-24\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-18±\sqrt{324+12\left(-24\right)}}{2\left(-3\right)}$$

Multiply 12 times -24.

$$x=\frac{-18±\sqrt{324-288}}{2\left(-3\right)}$$

Add 324 to -288.

$$x=\frac{-18±\sqrt{36}}{2\left(-3\right)}$$

Take the square root of 36.

$$x=\frac{-18±6}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-18±6}{-6}$$

Now solve the equation x=\frac{-18±6}{-6} when ± is plus. Add -18 to 6.

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

Divide -12 by -6.

$$x=2$$

Now solve the equation x=\frac{-18±6}{-6} when ± is minus. Subtract 6 from -18.

$$x=-\frac{24}{-6}$$

Divide -24 by -6.

$$x=4$$

The equation is now solved.

$$x=2 x=4$$

Steps for Completing the Square

Variable x cannot be equal to 3 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-3\right), the least common multiple of 3,x-3.

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

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

$$\left(3x-9\right)\times 2+\left(x-3\right)\left(x+4\right)=\left(x-3\right)\left(4x+4\right)+3\left(2-x\right)$$

Use the distributive property to multiply 3x-9 by 2.

$$6x-18+\left(x-3\right)\left(x+4\right)=\left(x-3\right)\left(4x+4\right)+3\left(2-x\right)$$

Use the distributive property to multiply x-3 by x+4 and combine like terms.

$$6x-18+x^{2}+x-12=\left(x-3\right)\left(4x+4\right)+3\left(2-x\right)$$

Combine 6x and x to get 7x.

$$7x-18+x^{2}-12=\left(x-3\right)\left(4x+4\right)+3\left(2-x\right)$$

Subtract 12 from -18 to get -30.

$$7x-30+x^{2}=\left(x-3\right)\left(4x+4\right)+3\left(2-x\right)$$

Use the distributive property to multiply x-3 by 4x+4 and combine like terms.

$$7x-30+x^{2}=4x^{2}-8x-12+3\left(2-x\right)$$

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

$$7x-30+x^{2}=4x^{2}-8x-12+6-3x$$

Add -12 and 6 to get -6.

$$7x-30+x^{2}=4x^{2}-8x-6-3x$$

Combine -8x and -3x to get -11x.

$$7x-30+x^{2}=4x^{2}-11x-6$$

Subtract 4x^{2} from both sides.

$$7x-30+x^{2}-4x^{2}=-11x-6$$

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

$$7x-30-3x^{2}=-11x-6$$

Add 11x to both sides.

$$7x-30-3x^{2}+11x=-6$$

Combine 7x and 11x to get 18x.

$$18x-30-3x^{2}=-6$$

Add 30 to both sides.

$$18x-3x^{2}=-6+30$$

Add -6 and 30 to get 24.

$$18x-3x^{2}=24$$

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.

$$-3x^{2}+18x=24$$

Divide both sides by -3.

$$\frac{-3x^{2}+18x}{-3}=\frac{24}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{18}{-3}x=\frac{24}{-3}$$

Divide 18 by -3.

$$x^{2}-6x=\frac{24}{-3}$$

Divide 24 by -3.

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

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=-8+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-8+9$$

Add -8 to 9.

$$x^{2}-6x+9=1$$

Factor x^{2}-6x+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}.

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

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{1}$$

Simplify.

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

Add 3 to both sides of the equation.

$$x=4 x=2$$