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

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

Solve for x (complex solution)

$$x=\frac{-7+\sqrt{239}i}{6}\approx -1.166666667+2.576604139i$$

$$x=\frac{-\sqrt{239}i-7}{6}\approx -1.166666667-2.576604139i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 10, the least common multiple of 5,2.

$$2\left(x+1\right)\left(x-2\right)-5\left(x-1\right)\left(x+2\right)=30$$

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

$$\left(2x+2\right)\left(x-2\right)-5\left(x-1\right)\left(x+2\right)=30$$

Use the distributive property to multiply 2x+2 by x-2 and combine like terms.

$$2x^{2}-2x-4-5\left(x-1\right)\left(x+2\right)=30$$

Use the distributive property to multiply -5 by x-1.

$$2x^{2}-2x-4+\left(-5x+5\right)\left(x+2\right)=30$$

Use the distributive property to multiply -5x+5 by x+2 and combine like terms.

$$2x^{2}-2x-4-5x^{2}-5x+10=30$$

Combine 2x^{2} and -5x^{2} to get -3x^{2}.

$$-3x^{2}-2x-4-5x+10=30$$

Combine -2x and -5x to get -7x.

$$-3x^{2}-7x-4+10=30$$

Add -4 and 10 to get 6.

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

Subtract 30 from both sides.

$$-3x^{2}-7x+6-30=0$$

Subtract 30 from 6 to get -24.

$$-3x^{2}-7x-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -7 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-3\right)\left(-24\right)}}{2\left(-3\right)}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-3\right)\left(-24\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-7\right)±\sqrt{49+12\left(-24\right)}}{2\left(-3\right)}$$

Multiply 12 times -24.

$$x=\frac{-\left(-7\right)±\sqrt{49-288}}{2\left(-3\right)}$$

Add 49 to -288.

$$x=\frac{-\left(-7\right)±\sqrt{-239}}{2\left(-3\right)}$$

Take the square root of -239.

$$x=\frac{-\left(-7\right)±\sqrt{239}i}{2\left(-3\right)}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{239}i}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{7±\sqrt{239}i}{-6}$$

Now solve the equation x=\frac{7±\sqrt{239}i}{-6} when ± is plus. Add 7 to i\sqrt{239}.

$$x=\frac{7+\sqrt{239}i}{-6}$$

Divide 7+i\sqrt{239} by -6.

$$x=\frac{-\sqrt{239}i-7}{6}$$

Now solve the equation x=\frac{7±\sqrt{239}i}{-6} when ± is minus. Subtract i\sqrt{239} from 7.

$$x=\frac{-\sqrt{239}i+7}{-6}$$

Divide 7-i\sqrt{239} by -6.

$$x=\frac{-7+\sqrt{239}i}{6}$$

The equation is now solved.

$$x=\frac{-\sqrt{239}i-7}{6} x=\frac{-7+\sqrt{239}i}{6}$$

Steps for Completing the Square

Multiply both sides of the equation by 10, the least common multiple of 5,2.

$$2\left(x+1\right)\left(x-2\right)-5\left(x-1\right)\left(x+2\right)=30$$

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

$$\left(2x+2\right)\left(x-2\right)-5\left(x-1\right)\left(x+2\right)=30$$

Use the distributive property to multiply 2x+2 by x-2 and combine like terms.

$$2x^{2}-2x-4-5\left(x-1\right)\left(x+2\right)=30$$

Use the distributive property to multiply -5 by x-1.

$$2x^{2}-2x-4+\left(-5x+5\right)\left(x+2\right)=30$$

Use the distributive property to multiply -5x+5 by x+2 and combine like terms.

$$2x^{2}-2x-4-5x^{2}-5x+10=30$$

Combine 2x^{2} and -5x^{2} to get -3x^{2}.

$$-3x^{2}-2x-4-5x+10=30$$

Combine -2x and -5x to get -7x.

$$-3x^{2}-7x-4+10=30$$

Add -4 and 10 to get 6.

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

Subtract 6 from both sides.

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

Subtract 6 from 30 to get 24.

$$-3x^{2}-7x=24$$

Divide both sides by -3.

$$\frac{-3x^{2}-7x}{-3}=\frac{24}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{7}{-3}\right)x=\frac{24}{-3}$$

Divide -7 by -3.

$$x^{2}+\frac{7}{3}x=\frac{24}{-3}$$

Divide 24 by -3.

$$x^{2}+\frac{7}{3}x=-8$$

Divide \frac{7}{3}, the coefficient of the x term, by 2 to get \frac{7}{6}. Then add the square of \frac{7}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{3}x+\left(\frac{7}{6}\right)^{2}=-8+\left(\frac{7}{6}\right)^{2}$$

Square \frac{7}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=-8+\frac{49}{36}$$

Add -8 to \frac{49}{36}.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=-\frac{239}{36}$$

Factor x^{2}+\frac{7}{3}x+\frac{49}{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}.

$$\left(x+\frac{7}{6}\right)^{2}=-\frac{239}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{6}\right)^{2}}=\sqrt{-\frac{239}{36}}$$

Simplify.

$$x+\frac{7}{6}=\frac{\sqrt{239}i}{6} x+\frac{7}{6}=-\frac{\sqrt{239}i}{6}$$

Subtract \frac{7}{6} from both sides of the equation.

$$x=\frac{-7+\sqrt{239}i}{6} x=\frac{-\sqrt{239}i-7}{6}$$