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

$$( x - 2 ) ( 3 x - 1 ) = 3 ( x + 1 ) \times - 2 ( 5 x + 1 )$$

Solve for x (complex solution)

$$x=\frac{-29+\sqrt{215}i}{66}\approx -0.439393939+0.222164823i$$

$$x=\frac{-\sqrt{215}i-29}{66}\approx -0.439393939-0.222164823i$$

Steps Using the Quadratic Formula

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

$$3x^{2}-7x+2=3\left(x+1\right)\left(-2\right)\left(5x+1\right)$$

Multiply 3 and -2 to get -6.

$$3x^{2}-7x+2=-6\left(x+1\right)\left(5x+1\right)$$

Use the distributive property to multiply -6 by x+1.

$$3x^{2}-7x+2=\left(-6x-6\right)\left(5x+1\right)$$

Use the distributive property to multiply -6x-6 by 5x+1 and combine like terms.

$$3x^{2}-7x+2=-30x^{2}-36x-6$$

Add 30x^{2} to both sides.

$$3x^{2}-7x+2+30x^{2}=-36x-6$$

Combine 3x^{2} and 30x^{2} to get 33x^{2}.

$$33x^{2}-7x+2=-36x-6$$

Add 36x to both sides.

$$33x^{2}-7x+2+36x=-6$$

Combine -7x and 36x to get 29x.

$$33x^{2}+29x+2=-6$$

Add 6 to both sides.

$$33x^{2}+29x+2+6=0$$

Add 2 and 6 to get 8.

$$33x^{2}+29x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 33 for a, 29 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-29±\sqrt{29^{2}-4\times 33\times 8}}{2\times 33}$$

Square 29.

$$x=\frac{-29±\sqrt{841-4\times 33\times 8}}{2\times 33}$$

Multiply -4 times 33.

$$x=\frac{-29±\sqrt{841-132\times 8}}{2\times 33}$$

Multiply -132 times 8.

$$x=\frac{-29±\sqrt{841-1056}}{2\times 33}$$

Add 841 to -1056.

$$x=\frac{-29±\sqrt{-215}}{2\times 33}$$

Take the square root of -215.

$$x=\frac{-29±\sqrt{215}i}{2\times 33}$$

Multiply 2 times 33.

$$x=\frac{-29±\sqrt{215}i}{66}$$

Now solve the equation x=\frac{-29±\sqrt{215}i}{66} when ± is plus. Add -29 to i\sqrt{215}.

$$x=\frac{-29+\sqrt{215}i}{66}$$

Now solve the equation x=\frac{-29±\sqrt{215}i}{66} when ± is minus. Subtract i\sqrt{215} from -29.

$$x=\frac{-\sqrt{215}i-29}{66}$$

The equation is now solved.

$$x=\frac{-29+\sqrt{215}i}{66} x=\frac{-\sqrt{215}i-29}{66}$$

Steps for Completing the Square

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

$$3x^{2}-7x+2=3\left(x+1\right)\left(-2\right)\left(5x+1\right)$$

Multiply 3 and -2 to get -6.

$$3x^{2}-7x+2=-6\left(x+1\right)\left(5x+1\right)$$

Use the distributive property to multiply -6 by x+1.

$$3x^{2}-7x+2=\left(-6x-6\right)\left(5x+1\right)$$

Use the distributive property to multiply -6x-6 by 5x+1 and combine like terms.

$$3x^{2}-7x+2=-30x^{2}-36x-6$$

Add 30x^{2} to both sides.

$$3x^{2}-7x+2+30x^{2}=-36x-6$$

Combine 3x^{2} and 30x^{2} to get 33x^{2}.

$$33x^{2}-7x+2=-36x-6$$

Add 36x to both sides.

$$33x^{2}-7x+2+36x=-6$$

Combine -7x and 36x to get 29x.

$$33x^{2}+29x+2=-6$$

Subtract 2 from both sides.

$$33x^{2}+29x=-6-2$$

Subtract 2 from -6 to get -8.

$$33x^{2}+29x=-8$$

Divide both sides by 33.

$$\frac{33x^{2}+29x}{33}=-\frac{8}{33}$$

Dividing by 33 undoes the multiplication by 33.

$$x^{2}+\frac{29}{33}x=-\frac{8}{33}$$

Divide \frac{29}{33}, the coefficient of the x term, by 2 to get \frac{29}{66}. Then add the square of \frac{29}{66} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{29}{33}x+\left(\frac{29}{66}\right)^{2}=-\frac{8}{33}+\left(\frac{29}{66}\right)^{2}$$

Square \frac{29}{66} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{29}{33}x+\frac{841}{4356}=-\frac{8}{33}+\frac{841}{4356}$$

Add -\frac{8}{33} to \frac{841}{4356} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{29}{33}x+\frac{841}{4356}=-\frac{215}{4356}$$

Factor x^{2}+\frac{29}{33}x+\frac{841}{4356}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{29}{66}\right)^{2}=-\frac{215}{4356}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{29}{66}\right)^{2}}=\sqrt{-\frac{215}{4356}}$$

Simplify.

$$x+\frac{29}{66}=\frac{\sqrt{215}i}{66} x+\frac{29}{66}=-\frac{\sqrt{215}i}{66}$$

Subtract \frac{29}{66} from both sides of the equation.

$$x=\frac{-29+\sqrt{215}i}{66} x=\frac{-\sqrt{215}i-29}{66}$$