Solve 3/2x+6-3x/4-9-6x/4=5x-11/2+3-frac{1-x}{2}x

$$\frac { 3 } { 2 } x + \frac { 6 - 3 x } { 4 } - \frac { 9 - 6 x } { 4 } = \frac { 5 x - 11 } { 2 } + 3 - \frac { 1 - x } { 2 } x$$

Solve for x

$$x = \frac{\sqrt{57} + 1}{4} \approx 2.137458609$$

$$x=\frac{1-\sqrt{57}}{4}\approx -1.637458609$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$6x+6-3x-\left(9-6x\right)=4\left(\frac{5x-11}{2}+3\right)-2\left(1-x\right)x$$

Combine 6x and -3x to get 3x.

$$3x+6-\left(9-6x\right)=4\left(\frac{5x-11}{2}+3\right)-2\left(1-x\right)x$$

To find the opposite of 9-6x, find the opposite of each term.

$$3x+6-9-\left(-6x\right)=4\left(\frac{5x-11}{2}+3\right)-2\left(1-x\right)x$$

The opposite of -6x is 6x.

$$3x+6-9+6x=4\left(\frac{5x-11}{2}+3\right)-2\left(1-x\right)x$$

Subtract 9 from 6 to get -3.

$$3x-3+6x=4\left(\frac{5x-11}{2}+3\right)-2\left(1-x\right)x$$

Combine 3x and 6x to get 9x.

$$9x-3=4\left(\frac{5x-11}{2}+3\right)-2\left(1-x\right)x$$

Use the distributive property to multiply 4 by \frac{5x-11}{2}+3.

$$9x-3=4\times \frac{5x-11}{2}+12-2\left(1-x\right)x$$

Cancel out 2, the greatest common factor in 4 and 2.

$$9x-3=2\left(5x-11\right)+12-2\left(1-x\right)x$$

Use the distributive property to multiply 2 by 5x-11.

$$9x-3=10x-22+12-2\left(1-x\right)x$$

Add -22 and 12 to get -10.

$$9x-3=10x-10-2\left(1-x\right)x$$

Add 2\left(1-x\right)x to both sides.

$$9x-3+2\left(1-x\right)x=10x-10$$

Use the distributive property to multiply 2 by 1-x.

$$9x-3+\left(2-2x\right)x=10x-10$$

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

$$9x-3+2x-2x^{2}=10x-10$$

Combine 9x and 2x to get 11x.

$$11x-3-2x^{2}=10x-10$$

Subtract 10x from both sides.

$$11x-3-2x^{2}-10x=-10$$

Combine 11x and -10x to get x.

$$x-3-2x^{2}=-10$$

Add 10 to both sides.

$$x-3-2x^{2}+10=0$$

Add -3 and 10 to get 7.

$$x+7-2x^{2}=0$$

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.

$$-2x^{2}+x+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 1 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-2\right)\times 7}}{2\left(-2\right)}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-2\right)\times 7}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-1±\sqrt{1+8\times 7}}{2\left(-2\right)}$$

Multiply 8 times 7.

$$x=\frac{-1±\sqrt{1+56}}{2\left(-2\right)}$$

Add 1 to 56.

$$x=\frac{-1±\sqrt{57}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-1±\sqrt{57}}{-4}$$

Now solve the equation x=\frac{-1±\sqrt{57}}{-4} when ± is plus. Add -1 to \sqrt{57}.

$$x=\frac{\sqrt{57}-1}{-4}$$

Divide -1+\sqrt{57} by -4.

$$x=\frac{1-\sqrt{57}}{4}$$

Now solve the equation x=\frac{-1±\sqrt{57}}{-4} when ± is minus. Subtract \sqrt{57} from -1.

$$x=\frac{-\sqrt{57}-1}{-4}$$

Divide -1-\sqrt{57} by -4.

$$x=\frac{\sqrt{57}+1}{4}$$

The equation is now solved.

$$x=\frac{1-\sqrt{57}}{4} x=\frac{\sqrt{57}+1}{4}$$

Steps for Completing the Square

Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$6x+6-3x-\left(9-6x\right)=4\left(\frac{5x-11}{2}+3\right)-2\left(1-x\right)x$$

Combine 6x and -3x to get 3x.

$$3x+6-\left(9-6x\right)=4\left(\frac{5x-11}{2}+3\right)-2\left(1-x\right)x$$

To find the opposite of 9-6x, find the opposite of each term.

$$3x+6-9-\left(-6x\right)=4\left(\frac{5x-11}{2}+3\right)-2\left(1-x\right)x$$

The opposite of -6x is 6x.

$$3x+6-9+6x=4\left(\frac{5x-11}{2}+3\right)-2\left(1-x\right)x$$

Subtract 9 from 6 to get -3.

$$3x-3+6x=4\left(\frac{5x-11}{2}+3\right)-2\left(1-x\right)x$$

Combine 3x and 6x to get 9x.

$$9x-3=4\left(\frac{5x-11}{2}+3\right)-2\left(1-x\right)x$$

Use the distributive property to multiply 4 by \frac{5x-11}{2}+3.

$$9x-3=4\times \frac{5x-11}{2}+12-2\left(1-x\right)x$$

Cancel out 2, the greatest common factor in 4 and 2.

$$9x-3=2\left(5x-11\right)+12-2\left(1-x\right)x$$

Use the distributive property to multiply 2 by 5x-11.

$$9x-3=10x-22+12-2\left(1-x\right)x$$

Add -22 and 12 to get -10.

$$9x-3=10x-10-2\left(1-x\right)x$$

Add 2\left(1-x\right)x to both sides.

$$9x-3+2\left(1-x\right)x=10x-10$$

Use the distributive property to multiply 2 by 1-x.

$$9x-3+\left(2-2x\right)x=10x-10$$

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

$$9x-3+2x-2x^{2}=10x-10$$

Combine 9x and 2x to get 11x.

$$11x-3-2x^{2}=10x-10$$

Subtract 10x from both sides.

$$11x-3-2x^{2}-10x=-10$$

Combine 11x and -10x to get x.

$$x-3-2x^{2}=-10$$

Add 3 to both sides.

$$x-2x^{2}=-10+3$$

Add -10 and 3 to get -7.

$$x-2x^{2}=-7$$

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.

$$-2x^{2}+x=-7$$

Divide both sides by -2.

$$\frac{-2x^{2}+x}{-2}=-\frac{7}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{1}{-2}x=-\frac{7}{-2}$$

Divide 1 by -2.

$$x^{2}-\frac{1}{2}x=-\frac{7}{-2}$$

Divide -7 by -2.

$$x^{2}-\frac{1}{2}x=\frac{7}{2}$$

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=\frac{7}{2}+\left(-\frac{1}{4}\right)^{2}$$

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{7}{2}+\frac{1}{16}$$

Add \frac{7}{2} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{57}{16}$$

Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{57}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{57}{16}}$$

Simplify.

$$x-\frac{1}{4}=\frac{\sqrt{57}}{4} x-\frac{1}{4}=-\frac{\sqrt{57}}{4}$$

Add \frac{1}{4} to both sides of the equation.

$$x=\frac{\sqrt{57}+1}{4} x=\frac{1-\sqrt{57}}{4}$$