Solve x=left(-2xright)^2left(3+3*7*2right)+3*2

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

Solve for x (complex solution)

$$x=\frac{-\sqrt{4319}i+1}{360}\approx 0.002777778-0.182553053i$$

$$x=\frac{1+\sqrt{4319}i}{360}\approx 0.002777778+0.182553053i$$

Steps Using the Quadratic Formula

Expand \left(-2x\right)^{2}.

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

Calculate -2 to the power of 2 and get 4.

$$x=4x^{2}\left(3+3\times 7\times 2\right)+3\times 2$$

Multiply 3 and 7 to get 21.

$$x=4x^{2}\left(3+21\times 2\right)+3\times 2$$

Multiply 21 and 2 to get 42.

$$x=4x^{2}\left(3+42\right)+3\times 2$$

Add 3 and 42 to get 45.

$$x=4x^{2}\times 45+3\times 2$$

Multiply 4 and 45 to get 180.

$$x=180x^{2}+3\times 2$$

Multiply 3 and 2 to get 6.

$$x=180x^{2}+6$$

Subtract 180x^{2} from both sides.

$$x-180x^{2}=6$$

Subtract 6 from both sides.

$$x-180x^{2}-6=0$$

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.

$$-180x^{2}+x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -180 for a, 1 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-180\right)\left(-6\right)}}{2\left(-180\right)}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-180\right)\left(-6\right)}}{2\left(-180\right)}$$

Multiply -4 times -180.

$$x=\frac{-1±\sqrt{1+720\left(-6\right)}}{2\left(-180\right)}$$

Multiply 720 times -6.

$$x=\frac{-1±\sqrt{1-4320}}{2\left(-180\right)}$$

Add 1 to -4320.

$$x=\frac{-1±\sqrt{-4319}}{2\left(-180\right)}$$

Take the square root of -4319.

$$x=\frac{-1±\sqrt{4319}i}{2\left(-180\right)}$$

Multiply 2 times -180.

$$x=\frac{-1±\sqrt{4319}i}{-360}$$

Now solve the equation x=\frac{-1±\sqrt{4319}i}{-360} when ± is plus. Add -1 to i\sqrt{4319}.

$$x=\frac{-1+\sqrt{4319}i}{-360}$$

Divide -1+i\sqrt{4319} by -360.

$$x=\frac{-\sqrt{4319}i+1}{360}$$

Now solve the equation x=\frac{-1±\sqrt{4319}i}{-360} when ± is minus. Subtract i\sqrt{4319} from -1.

$$x=\frac{-\sqrt{4319}i-1}{-360}$$

Divide -1-i\sqrt{4319} by -360.

$$x=\frac{1+\sqrt{4319}i}{360}$$

The equation is now solved.

$$x=\frac{-\sqrt{4319}i+1}{360} x=\frac{1+\sqrt{4319}i}{360}$$

Steps for Completing the Square

Expand \left(-2x\right)^{2}.

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

Calculate -2 to the power of 2 and get 4.

$$x=4x^{2}\left(3+3\times 7\times 2\right)+3\times 2$$

Multiply 3 and 7 to get 21.

$$x=4x^{2}\left(3+21\times 2\right)+3\times 2$$

Multiply 21 and 2 to get 42.

$$x=4x^{2}\left(3+42\right)+3\times 2$$

Add 3 and 42 to get 45.

$$x=4x^{2}\times 45+3\times 2$$

Multiply 4 and 45 to get 180.

$$x=180x^{2}+3\times 2$$

Multiply 3 and 2 to get 6.

$$x=180x^{2}+6$$

Subtract 180x^{2} from both sides.

$$x-180x^{2}=6$$

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.

$$-180x^{2}+x=6$$

Divide both sides by -180.

$$\frac{-180x^{2}+x}{-180}=\frac{6}{-180}$$

Dividing by -180 undoes the multiplication by -180.

$$x^{2}+\frac{1}{-180}x=\frac{6}{-180}$$

Divide 1 by -180.

$$x^{2}-\frac{1}{180}x=\frac{6}{-180}$$

Reduce the fraction \frac{6}{-180} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{1}{180}x=-\frac{1}{30}$$

Divide -\frac{1}{180}, the coefficient of the x term, by 2 to get -\frac{1}{360}. Then add the square of -\frac{1}{360} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{180}x+\left(-\frac{1}{360}\right)^{2}=-\frac{1}{30}+\left(-\frac{1}{360}\right)^{2}$$

Square -\frac{1}{360} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{180}x+\frac{1}{129600}=-\frac{1}{30}+\frac{1}{129600}$$

Add -\frac{1}{30} to \frac{1}{129600} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{180}x+\frac{1}{129600}=-\frac{4319}{129600}$$

Factor x^{2}-\frac{1}{180}x+\frac{1}{129600}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{360}\right)^{2}=-\frac{4319}{129600}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{360}\right)^{2}}=\sqrt{-\frac{4319}{129600}}$$

Simplify.

$$x-\frac{1}{360}=\frac{\sqrt{4319}i}{360} x-\frac{1}{360}=-\frac{\sqrt{4319}i}{360}$$

Add \frac{1}{360} to both sides of the equation.

$$x=\frac{1+\sqrt{4319}i}{360} x=\frac{-\sqrt{4319}i+1}{360}$$