Solve -9/-2+6x-5/2-3x=2x-2

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

Solve for x

$$x = \frac{\sqrt{457} + 35}{24} \approx 2.34906493$$

$$x=\frac{35-\sqrt{457}}{24}\approx 0.567601736$$

Steps Using the Quadratic Formula

Variable x cannot be equal to \frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by 2\left(3x-2\right), the least common multiple of -2,2-3x.

$$\left(-3x+2\right)\left(-9\right)-2\left(6x-5\right)=2x\times 2\left(3x-2\right)+2\left(3x-2\right)\left(-2\right)$$

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

$$27x-18-2\left(6x-5\right)=2x\times 2\left(3x-2\right)+2\left(3x-2\right)\left(-2\right)$$

Use the distributive property to multiply -2 by 6x-5.

$$27x-18-12x+10=2x\times 2\left(3x-2\right)+2\left(3x-2\right)\left(-2\right)$$

Combine 27x and -12x to get 15x.

$$15x-18+10=2x\times 2\left(3x-2\right)+2\left(3x-2\right)\left(-2\right)$$

Add -18 and 10 to get -8.

$$15x-8=2x\times 2\left(3x-2\right)+2\left(3x-2\right)\left(-2\right)$$

Multiply 2 and 2 to get 4.

$$15x-8=4x\left(3x-2\right)+2\left(3x-2\right)\left(-2\right)$$

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

$$15x-8=12x^{2}-8x+2\left(3x-2\right)\left(-2\right)$$

Multiply 2 and -2 to get -4.

$$15x-8=12x^{2}-8x-4\left(3x-2\right)$$

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

$$15x-8=12x^{2}-8x-12x+8$$

Combine -8x and -12x to get -20x.

$$15x-8=12x^{2}-20x+8$$

Subtract 12x^{2} from both sides.

$$15x-8-12x^{2}=-20x+8$$

Add 20x to both sides.

$$15x-8-12x^{2}+20x=8$$

Combine 15x and 20x to get 35x.

$$35x-8-12x^{2}=8$$

Subtract 8 from both sides.

$$35x-8-12x^{2}-8=0$$

Subtract 8 from -8 to get -16.

$$35x-16-12x^{2}=0$$

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.

$$-12x^{2}+35x-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -12 for a, 35 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-35±\sqrt{35^{2}-4\left(-12\right)\left(-16\right)}}{2\left(-12\right)}$$

Square 35.

$$x=\frac{-35±\sqrt{1225-4\left(-12\right)\left(-16\right)}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$x=\frac{-35±\sqrt{1225+48\left(-16\right)}}{2\left(-12\right)}$$

Multiply 48 times -16.

$$x=\frac{-35±\sqrt{1225-768}}{2\left(-12\right)}$$

Add 1225 to -768.

$$x=\frac{-35±\sqrt{457}}{2\left(-12\right)}$$

Multiply 2 times -12.

$$x=\frac{-35±\sqrt{457}}{-24}$$

Now solve the equation x=\frac{-35±\sqrt{457}}{-24} when ± is plus. Add -35 to \sqrt{457}.

$$x=\frac{\sqrt{457}-35}{-24}$$

Divide -35+\sqrt{457} by -24.

$$x=\frac{35-\sqrt{457}}{24}$$

Now solve the equation x=\frac{-35±\sqrt{457}}{-24} when ± is minus. Subtract \sqrt{457} from -35.

$$x=\frac{-\sqrt{457}-35}{-24}$$

Divide -35-\sqrt{457} by -24.

$$x=\frac{\sqrt{457}+35}{24}$$

The equation is now solved.

$$x=\frac{35-\sqrt{457}}{24} x=\frac{\sqrt{457}+35}{24}$$

Steps for Completing the Square

Variable x cannot be equal to \frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by 2\left(3x-2\right), the least common multiple of -2,2-3x.

$$\left(-3x+2\right)\left(-9\right)-2\left(6x-5\right)=2x\times 2\left(3x-2\right)+2\left(3x-2\right)\left(-2\right)$$

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

$$27x-18-2\left(6x-5\right)=2x\times 2\left(3x-2\right)+2\left(3x-2\right)\left(-2\right)$$

Use the distributive property to multiply -2 by 6x-5.

$$27x-18-12x+10=2x\times 2\left(3x-2\right)+2\left(3x-2\right)\left(-2\right)$$

Combine 27x and -12x to get 15x.

$$15x-18+10=2x\times 2\left(3x-2\right)+2\left(3x-2\right)\left(-2\right)$$

Add -18 and 10 to get -8.

$$15x-8=2x\times 2\left(3x-2\right)+2\left(3x-2\right)\left(-2\right)$$

Multiply 2 and 2 to get 4.

$$15x-8=4x\left(3x-2\right)+2\left(3x-2\right)\left(-2\right)$$

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

$$15x-8=12x^{2}-8x+2\left(3x-2\right)\left(-2\right)$$

Multiply 2 and -2 to get -4.

$$15x-8=12x^{2}-8x-4\left(3x-2\right)$$

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

$$15x-8=12x^{2}-8x-12x+8$$

Combine -8x and -12x to get -20x.

$$15x-8=12x^{2}-20x+8$$

Subtract 12x^{2} from both sides.

$$15x-8-12x^{2}=-20x+8$$

Add 20x to both sides.

$$15x-8-12x^{2}+20x=8$$

Combine 15x and 20x to get 35x.

$$35x-8-12x^{2}=8$$

Add 8 to both sides.

$$35x-12x^{2}=8+8$$

Add 8 and 8 to get 16.

$$35x-12x^{2}=16$$

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.

$$-12x^{2}+35x=16$$

Divide both sides by -12.

$$\frac{-12x^{2}+35x}{-12}=\frac{16}{-12}$$

Dividing by -12 undoes the multiplication by -12.

$$x^{2}+\frac{35}{-12}x=\frac{16}{-12}$$

Divide 35 by -12.

$$x^{2}-\frac{35}{12}x=\frac{16}{-12}$$

Reduce the fraction \frac{16}{-12} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{35}{12}x=-\frac{4}{3}$$

Divide -\frac{35}{12}, the coefficient of the x term, by 2 to get -\frac{35}{24}. Then add the square of -\frac{35}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{35}{12}x+\left(-\frac{35}{24}\right)^{2}=-\frac{4}{3}+\left(-\frac{35}{24}\right)^{2}$$

Square -\frac{35}{24} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{35}{12}x+\frac{1225}{576}=-\frac{4}{3}+\frac{1225}{576}$$

Add -\frac{4}{3} to \frac{1225}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{35}{12}x+\frac{1225}{576}=\frac{457}{576}$$

Factor x^{2}-\frac{35}{12}x+\frac{1225}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{35}{24}\right)^{2}=\frac{457}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{35}{24}\right)^{2}}=\sqrt{\frac{457}{576}}$$

Simplify.

$$x-\frac{35}{24}=\frac{\sqrt{457}}{24} x-\frac{35}{24}=-\frac{\sqrt{457}}{24}$$

Add \frac{35}{24} to both sides of the equation.

$$x=\frac{\sqrt{457}+35}{24} x=\frac{35-\sqrt{457}}{24}$$