Solve (x-2)(1-3x)=6

$$(x-2)(1-3x)=6$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{47}i+7}{6}\approx 1.166666667-1.1426091i$$

$$x=\frac{7+\sqrt{47}i}{6}\approx 1.166666667+1.1426091i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-2 by 1-3x and combine like terms.

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

Subtract 6 from both sides.

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

Subtract 6 from -2 to get -8.

$$7x-3x^{2}-8=0$$

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.

$$-3x^{2}+7x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 7 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-3\right)\left(-8\right)}}{2\left(-3\right)}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-3\right)\left(-8\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-7±\sqrt{49+12\left(-8\right)}}{2\left(-3\right)}$$

Multiply 12 times -8.

$$x=\frac{-7±\sqrt{49-96}}{2\left(-3\right)}$$

Add 49 to -96.

$$x=\frac{-7±\sqrt{-47}}{2\left(-3\right)}$$

Take the square root of -47.

$$x=\frac{-7±\sqrt{47}i}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-7±\sqrt{47}i}{-6}$$

Now solve the equation x=\frac{-7±\sqrt{47}i}{-6} when ± is plus. Add -7 to i\sqrt{47}.

$$x=\frac{-7+\sqrt{47}i}{-6}$$

Divide -7+i\sqrt{47} by -6.

$$x=\frac{-\sqrt{47}i+7}{6}$$

Now solve the equation x=\frac{-7±\sqrt{47}i}{-6} when ± is minus. Subtract i\sqrt{47} from -7.

$$x=\frac{-\sqrt{47}i-7}{-6}$$

Divide -7-i\sqrt{47} by -6.

$$x=\frac{7+\sqrt{47}i}{6}$$

The equation is now solved.

$$x=\frac{-\sqrt{47}i+7}{6} x=\frac{7+\sqrt{47}i}{6}$$

Steps for Completing the Square

Use the distributive property to multiply x-2 by 1-3x and combine like terms.

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

Add 2 to both sides.

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

Add 6 and 2 to get 8.

$$7x-3x^{2}=8$$

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.

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

Divide both sides by -3.

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

Dividing by -3 undoes the multiplication by -3.

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

Divide 7 by -3.

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

Divide 8 by -3.

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

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.

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

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

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

Add -\frac{8}{3} to \frac{49}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=-\frac{47}{36}$$

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}.

$$\left(x-\frac{7}{6}\right)^{2}=-\frac{47}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{6}\right)^{2}}=\sqrt{-\frac{47}{36}}$$

Simplify.

$$x-\frac{7}{6}=\frac{\sqrt{47}i}{6} x-\frac{7}{6}=-\frac{\sqrt{47}i}{6}$$

Add \frac{7}{6} to both sides of the equation.

$$x=\frac{7+\sqrt{47}i}{6} x=\frac{-\sqrt{47}i+7}{6}$$