Solve 6*(x+3(x-2)-(x+1)^2)=1

$$6 \cdot ( x + 3 ( x - 2 ) - ( x + 1 ) ^ { 2 } ) = 1$$

Solve for x (complex solution)

$$x=\frac{\sqrt{222}i}{6}+1\approx 1+2.483277404i$$

$$x=-\frac{\sqrt{222}i}{6}+1\approx 1-2.483277404i$$

Steps Using the Quadratic Formula

Divide both sides by 6.

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

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

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

Combine x and 3x to get 4x.

$$4x-6-\left(x+1\right)^{2}=\frac{1}{6}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$4x-6-\left(x^{2}+2x+1\right)=\frac{1}{6}$$

To find the opposite of x^{2}+2x+1, find the opposite of each term.

$$4x-6-x^{2}-2x-1=\frac{1}{6}$$

Combine 4x and -2x to get 2x.

$$2x-6-x^{2}-1=\frac{1}{6}$$

Subtract 1 from -6 to get -7.

$$2x-7-x^{2}=\frac{1}{6}$$

Subtract \frac{1}{6} from both sides.

$$2x-7-x^{2}-\frac{1}{6}=0$$

Subtract \frac{1}{6} from -7 to get -\frac{43}{6}.

$$2x-\frac{43}{6}-x^{2}=0$$

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.

$$-x^{2}+2x-\frac{43}{6}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and -\frac{43}{6} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\left(-\frac{43}{6}\right)}}{2\left(-1\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-1\right)\left(-\frac{43}{6}\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-2±\sqrt{4+4\left(-\frac{43}{6}\right)}}{2\left(-1\right)}$$

Multiply 4 times -\frac{43}{6}.

$$x=\frac{-2±\sqrt{4-\frac{86}{3}}}{2\left(-1\right)}$$

Add 4 to -\frac{86}{3}.

$$x=\frac{-2±\sqrt{-\frac{74}{3}}}{2\left(-1\right)}$$

Take the square root of -\frac{74}{3}.

$$x=\frac{-2±\frac{\sqrt{222}i}{3}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-2±\frac{\sqrt{222}i}{3}}{-2}$$

Now solve the equation x=\frac{-2±\frac{\sqrt{222}i}{3}}{-2} when ± is plus. Add -2 to \frac{i\sqrt{222}}{3}.

$$x=\frac{\frac{\sqrt{222}i}{3}-2}{-2}$$

Divide -2+\frac{i\sqrt{222}}{3} by -2.

$$x=-\frac{\sqrt{222}i}{6}+1$$

Now solve the equation x=\frac{-2±\frac{\sqrt{222}i}{3}}{-2} when ± is minus. Subtract \frac{i\sqrt{222}}{3} from -2.

$$x=\frac{-\frac{\sqrt{222}i}{3}-2}{-2}$$

Divide -2-\frac{i\sqrt{222}}{3} by -2.

$$x=\frac{\sqrt{222}i}{6}+1$$

The equation is now solved.

$$x=-\frac{\sqrt{222}i}{6}+1 x=\frac{\sqrt{222}i}{6}+1$$

Steps for Completing the Square

Divide both sides by 6.

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

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

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

Combine x and 3x to get 4x.

$$4x-6-\left(x+1\right)^{2}=\frac{1}{6}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$4x-6-\left(x^{2}+2x+1\right)=\frac{1}{6}$$

To find the opposite of x^{2}+2x+1, find the opposite of each term.

$$4x-6-x^{2}-2x-1=\frac{1}{6}$$

Combine 4x and -2x to get 2x.

$$2x-6-x^{2}-1=\frac{1}{6}$$

Subtract 1 from -6 to get -7.

$$2x-7-x^{2}=\frac{1}{6}$$

Add 7 to both sides.

$$2x-x^{2}=\frac{1}{6}+7$$

Add \frac{1}{6} and 7 to get \frac{43}{6}.

$$2x-x^{2}=\frac{43}{6}$$

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.

$$-x^{2}+2x=\frac{43}{6}$$

Divide both sides by -1.

$$\frac{-x^{2}+2x}{-1}=\frac{\frac{43}{6}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{2}{-1}x=\frac{\frac{43}{6}}{-1}$$

Divide 2 by -1.

$$x^{2}-2x=\frac{\frac{43}{6}}{-1}$$

Divide \frac{43}{6} by -1.

$$x^{2}-2x=-\frac{43}{6}$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=-\frac{43}{6}+1$$

Add -\frac{43}{6} to 1.

$$x^{2}-2x+1=-\frac{37}{6}$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=-\frac{37}{6}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-\frac{37}{6}}$$

Simplify.

$$x-1=\frac{\sqrt{222}i}{6} x-1=-\frac{\sqrt{222}i}{6}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{222}i}{6}+1 x=-\frac{\sqrt{222}i}{6}+1$$