Solve 3(-x+2)/5+4x(-2x+1/3)=x(-3x+1)-1/2

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

Solve for x (complex solution)

$$x=\frac{\sqrt{494}i}{10}+\frac{2}{5}\approx 0.4+2.222611077i$$

$$x=-\frac{\sqrt{494}i}{10}+\frac{2}{5}\approx 0.4-2.222611077i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 30, the least common multiple of 5,3,2.

$$6\times 3\left(-x+2\right)+40x\left(-2x+1\right)=30x\left(-3x+1\right)-15$$

Multiply 6 and 3 to get 18.

$$18\left(-x+2\right)+40x\left(-2x+1\right)=30x\left(-3x+1\right)-15$$

Use the distributive property to multiply 18 by -x+2.

$$18\left(-x\right)+36+40x\left(-2x+1\right)=30x\left(-3x+1\right)-15$$

Use the distributive property to multiply 40x by -2x+1.

$$18\left(-x\right)+36-80x^{2}+40x=30x\left(-3x+1\right)-15$$

Use the distributive property to multiply 30x by -3x+1.

$$18\left(-x\right)+36-80x^{2}+40x=-90x^{2}+30x-15$$

Add 90x^{2} to both sides.

$$18\left(-x\right)+36-80x^{2}+40x+90x^{2}=30x-15$$

Combine -80x^{2} and 90x^{2} to get 10x^{2}.

$$18\left(-x\right)+36+10x^{2}+40x=30x-15$$

Subtract 30x from both sides.

$$18\left(-x\right)+36+10x^{2}+40x-30x=-15$$

Combine 40x and -30x to get 10x.

$$18\left(-x\right)+36+10x^{2}+10x=-15$$

Add 15 to both sides.

$$18\left(-x\right)+36+10x^{2}+10x+15=0$$

Add 36 and 15 to get 51.

$$18\left(-x\right)+51+10x^{2}+10x=0$$

Multiply 18 and -1 to get -18.

$$-18x+51+10x^{2}+10x=0$$

Combine -18x and 10x to get -8x.

$$-8x+51+10x^{2}=0$$

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.

$$10x^{2}-8x+51=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, -8 for b, and 51 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 10\times 51}}{2\times 10}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 10\times 51}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-\left(-8\right)±\sqrt{64-40\times 51}}{2\times 10}$$

Multiply -40 times 51.

$$x=\frac{-\left(-8\right)±\sqrt{64-2040}}{2\times 10}$$

Add 64 to -2040.

$$x=\frac{-\left(-8\right)±\sqrt{-1976}}{2\times 10}$$

Take the square root of -1976.

$$x=\frac{-\left(-8\right)±2\sqrt{494}i}{2\times 10}$$

The opposite of -8 is 8.

$$x=\frac{8±2\sqrt{494}i}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{8±2\sqrt{494}i}{20}$$

Now solve the equation x=\frac{8±2\sqrt{494}i}{20} when ± is plus. Add 8 to 2i\sqrt{494}.

$$x=\frac{8+2\sqrt{494}i}{20}$$

Divide 8+2i\sqrt{494} by 20.

$$x=\frac{\sqrt{494}i}{10}+\frac{2}{5}$$

Now solve the equation x=\frac{8±2\sqrt{494}i}{20} when ± is minus. Subtract 2i\sqrt{494} from 8.

$$x=\frac{-2\sqrt{494}i+8}{20}$$

Divide 8-2i\sqrt{494} by 20.

$$x=-\frac{\sqrt{494}i}{10}+\frac{2}{5}$$

The equation is now solved.

$$x=\frac{\sqrt{494}i}{10}+\frac{2}{5} x=-\frac{\sqrt{494}i}{10}+\frac{2}{5}$$

Steps for Completing the Square

Multiply both sides of the equation by 30, the least common multiple of 5,3,2.

$$6\times 3\left(-x+2\right)+40x\left(-2x+1\right)=30x\left(-3x+1\right)-15$$

Multiply 6 and 3 to get 18.

$$18\left(-x+2\right)+40x\left(-2x+1\right)=30x\left(-3x+1\right)-15$$

Use the distributive property to multiply 18 by -x+2.

$$18\left(-x\right)+36+40x\left(-2x+1\right)=30x\left(-3x+1\right)-15$$

Use the distributive property to multiply 40x by -2x+1.

$$18\left(-x\right)+36-80x^{2}+40x=30x\left(-3x+1\right)-15$$

Use the distributive property to multiply 30x by -3x+1.

$$18\left(-x\right)+36-80x^{2}+40x=-90x^{2}+30x-15$$

Add 90x^{2} to both sides.

$$18\left(-x\right)+36-80x^{2}+40x+90x^{2}=30x-15$$

Combine -80x^{2} and 90x^{2} to get 10x^{2}.

$$18\left(-x\right)+36+10x^{2}+40x=30x-15$$

Subtract 30x from both sides.

$$18\left(-x\right)+36+10x^{2}+40x-30x=-15$$

Combine 40x and -30x to get 10x.

$$18\left(-x\right)+36+10x^{2}+10x=-15$$

Subtract 36 from both sides.

$$18\left(-x\right)+10x^{2}+10x=-15-36$$

Subtract 36 from -15 to get -51.

$$18\left(-x\right)+10x^{2}+10x=-51$$

Multiply 18 and -1 to get -18.

$$-18x+10x^{2}+10x=-51$$

Combine -18x and 10x to get -8x.

$$-8x+10x^{2}=-51$$

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.

$$10x^{2}-8x=-51$$

Divide both sides by 10.

$$\frac{10x^{2}-8x}{10}=-\frac{51}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\left(-\frac{8}{10}\right)x=-\frac{51}{10}$$

Reduce the fraction \frac{-8}{10} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{4}{5}x=-\frac{51}{10}$$

Divide -\frac{4}{5}, the coefficient of the x term, by 2 to get -\frac{2}{5}. Then add the square of -\frac{2}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{4}{5}x+\left(-\frac{2}{5}\right)^{2}=-\frac{51}{10}+\left(-\frac{2}{5}\right)^{2}$$

Square -\frac{2}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{4}{5}x+\frac{4}{25}=-\frac{51}{10}+\frac{4}{25}$$

Add -\frac{51}{10} to \frac{4}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{4}{5}x+\frac{4}{25}=-\frac{247}{50}$$

Factor x^{2}-\frac{4}{5}x+\frac{4}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2}{5}\right)^{2}=-\frac{247}{50}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2}{5}\right)^{2}}=\sqrt{-\frac{247}{50}}$$

Simplify.

$$x-\frac{2}{5}=\frac{\sqrt{494}i}{10} x-\frac{2}{5}=-\frac{\sqrt{494}i}{10}$$

Add \frac{2}{5} to both sides of the equation.

$$x=\frac{\sqrt{494}i}{10}+\frac{2}{5} x=-\frac{\sqrt{494}i}{10}+\frac{2}{5}$$