Solve x^2-1.2x+x^2+1.2^2=x^2

$${ x }^{ 2 } -1.2x+ { x }^{ 2 } + { 1.2 }^{ 2 } = { x }^{ 2 }$$

Solve for x (complex solution)

$$x=\frac{3+i\times 3\sqrt{3}}{5}\approx 0.6+1.039230485i$$

$$x=\frac{-i\times 3\sqrt{3}+3}{5}\approx 0.6-1.039230485i$$

Steps Using the Quadratic Formula

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

$$2x^{2}-1.2x+1.2^{2}=x^{2}$$

Calculate 1.2 to the power of 2 and get 1.44.

$$2x^{2}-1.2x+1.44=x^{2}$$

Subtract x^{2} from both sides.

$$2x^{2}-1.2x+1.44-x^{2}=0$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}-1.2x+1.44=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1.2 for b, and 1.44 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1.2\right)±\sqrt{\left(-1.2\right)^{2}-4\times 1.44}}{2}$$

Square -1.2 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-1.2\right)±\sqrt{1.44-4\times 1.44}}{2}$$

Multiply -4 times 1.44.

$$x=\frac{-\left(-1.2\right)±\sqrt{\frac{36-144}{25}}}{2}$$

Add 1.44 to -5.76 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-1.2\right)±\sqrt{-4.32}}{2}$$

Take the square root of -4.32.

$$x=\frac{-\left(-1.2\right)±\frac{6\sqrt{3}i}{5}}{2}$$

The opposite of -1.2 is 1.2.

$$x=\frac{1.2±\frac{6\sqrt{3}i}{5}}{2}$$

Now solve the equation x=\frac{1.2±\frac{6\sqrt{3}i}{5}}{2} when ± is plus. Add 1.2 to \frac{6i\sqrt{3}}{5}.

$$x=\frac{6+6\sqrt{3}i}{2\times 5}$$

Divide \frac{6+6i\sqrt{3}}{5} by 2.

$$x=\frac{3+3\sqrt{3}i}{5}$$

Now solve the equation x=\frac{1.2±\frac{6\sqrt{3}i}{5}}{2} when ± is minus. Subtract \frac{6i\sqrt{3}}{5} from 1.2.

$$x=\frac{-6\sqrt{3}i+6}{2\times 5}$$

Divide \frac{6-6i\sqrt{3}}{5} by 2.

$$x=\frac{-3\sqrt{3}i+3}{5}$$

The equation is now solved.

$$x=\frac{3+3\sqrt{3}i}{5} x=\frac{-3\sqrt{3}i+3}{5}$$

Steps for Completing the Square

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

$$2x^{2}-1.2x+1.2^{2}=x^{2}$$

Calculate 1.2 to the power of 2 and get 1.44.

$$2x^{2}-1.2x+1.44=x^{2}$$

Subtract x^{2} from both sides.

$$2x^{2}-1.2x+1.44-x^{2}=0$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}-1.2x+1.44=0$$

Subtract 1.44 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-1.2x=-1.44$$

Divide -1.2, the coefficient of the x term, by 2 to get -0.6. Then add the square of -0.6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-1.2x+\left(-0.6\right)^{2}=-1.44+\left(-0.6\right)^{2}$$

Square -0.6 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-1.2x+0.36=\frac{-36+9}{25}$$

Add -1.44 to 0.36 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-1.2x+0.36=-1.08$$

Factor x^{2}-1.2x+0.36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0.6\right)^{2}=-1.08$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-0.6\right)^{2}}=\sqrt{-1.08}$$

Simplify.

$$x-0.6=\frac{3\sqrt{3}i}{5} x-0.6=-\frac{3\sqrt{3}i}{5}$$

Add 0.6 to both sides of the equation.

$$x=\frac{3+3\sqrt{3}i}{5} x=\frac{-3\sqrt{3}i+3}{5}$$