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

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

Solve for x (complex solution)

$$x=\frac{-1+\sqrt{7}i}{2}\approx -0.5+1.322875656i$$

$$x=\frac{-\sqrt{7}i-1}{2}\approx -0.5-1.322875656i$$

Steps Using the Quadratic Formula

To find the opposite of x-1, find the opposite of each term.

$$x-\left(x-2\left(3x-x-\left(-1\right)\right)\right)=\left(-x\right)\left(x-3\right)$$

The opposite of -1 is 1.

$$x-\left(x-2\left(3x-x+1\right)\right)=\left(-x\right)\left(x-3\right)$$

Combine 3x and -x to get 2x.

$$x-\left(x-2\left(2x+1\right)\right)=\left(-x\right)\left(x-3\right)$$

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

$$x-\left(x-4x-2\right)=\left(-x\right)\left(x-3\right)$$

Combine x and -4x to get -3x.

$$x-\left(-3x-2\right)=\left(-x\right)\left(x-3\right)$$

To find the opposite of -3x-2, find the opposite of each term.

$$x-\left(-3x\right)-\left(-2\right)=\left(-x\right)\left(x-3\right)$$

The opposite of -3x is 3x.

$$x+3x-\left(-2\right)=\left(-x\right)\left(x-3\right)$$

The opposite of -2 is 2.

$$x+3x+2=\left(-x\right)\left(x-3\right)$$

Combine x and 3x to get 4x.

$$4x+2=\left(-x\right)\left(x-3\right)$$

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

$$4x+2=\left(-x\right)x-3\left(-x\right)$$

Multiply -3 and -1 to get 3.

$$4x+2=\left(-x\right)x+3x$$

Subtract \left(-x\right)x from both sides.

$$4x+2-\left(-x\right)x=3x$$

Subtract 3x from both sides.

$$4x+2-\left(-x\right)x-3x=0$$

Multiply x and x to get x^{2}.

$$4x+2-\left(-x^{2}\right)-3x=0$$

Multiply -1 and -1 to get 1.

$$4x+2+x^{2}-3x=0$$

Combine 4x and -3x to get x.

$$x+2+x^{2}=0$$

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.

$$x^{2}+x+2=0$$

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

$$x=\frac{-1±\sqrt{1^{2}-4\times 2}}{2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 2}}{2}$$

Multiply -4 times 2.

$$x=\frac{-1±\sqrt{1-8}}{2}$$

Add 1 to -8.

$$x=\frac{-1±\sqrt{-7}}{2}$$

Take the square root of -7.

$$x=\frac{-1±\sqrt{7}i}{2}$$

Now solve the equation x=\frac{-1±\sqrt{7}i}{2} when ± is plus. Add -1 to i\sqrt{7}.

$$x=\frac{-1+\sqrt{7}i}{2}$$

Now solve the equation x=\frac{-1±\sqrt{7}i}{2} when ± is minus. Subtract i\sqrt{7} from -1.

$$x=\frac{-\sqrt{7}i-1}{2}$$

The equation is now solved.

$$x=\frac{-1+\sqrt{7}i}{2} x=\frac{-\sqrt{7}i-1}{2}$$

Steps for Completing the Square

To find the opposite of x-1, find the opposite of each term.

$$x-\left(x-2\left(3x-x-\left(-1\right)\right)\right)=\left(-x\right)\left(x-3\right)$$

The opposite of -1 is 1.

$$x-\left(x-2\left(3x-x+1\right)\right)=\left(-x\right)\left(x-3\right)$$

Combine 3x and -x to get 2x.

$$x-\left(x-2\left(2x+1\right)\right)=\left(-x\right)\left(x-3\right)$$

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

$$x-\left(x-4x-2\right)=\left(-x\right)\left(x-3\right)$$

Combine x and -4x to get -3x.

$$x-\left(-3x-2\right)=\left(-x\right)\left(x-3\right)$$

To find the opposite of -3x-2, find the opposite of each term.

$$x-\left(-3x\right)-\left(-2\right)=\left(-x\right)\left(x-3\right)$$

The opposite of -3x is 3x.

$$x+3x-\left(-2\right)=\left(-x\right)\left(x-3\right)$$

The opposite of -2 is 2.

$$x+3x+2=\left(-x\right)\left(x-3\right)$$

Combine x and 3x to get 4x.

$$4x+2=\left(-x\right)\left(x-3\right)$$

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

$$4x+2=\left(-x\right)x-3\left(-x\right)$$

Multiply -3 and -1 to get 3.

$$4x+2=\left(-x\right)x+3x$$

Subtract \left(-x\right)x from both sides.

$$4x+2-\left(-x\right)x=3x$$

Subtract 3x from both sides.

$$4x+2-\left(-x\right)x-3x=0$$

Multiply x and x to get x^{2}.

$$4x+2-\left(-x^{2}\right)-3x=0$$

Multiply -1 and -1 to get 1.

$$4x+2+x^{2}-3x=0$$

Combine 4x and -3x to get x.

$$x+2+x^{2}=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$x+x^{2}=-2$$

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.

$$x^{2}+x=-2$$

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

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

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

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

Add -2 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=-\frac{7}{4}$$

Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{2}\right)^{2}=-\frac{7}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{-\frac{7}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{7}i}{2} x+\frac{1}{2}=-\frac{\sqrt{7}i}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{-1+\sqrt{7}i}{2} x=\frac{-\sqrt{7}i-1}{2}$$