Solve 2x^2=1.8x-1

$$2 { x }^{ 2 } =1.8x-1$$

Solve for x (complex solution)

$$x=\frac{9+i\sqrt{119}}{20}\approx 0.45+0.545435606i$$

$$x=\frac{-i\sqrt{119}+9}{20}\approx 0.45-0.545435606i$$

Steps Using the Quadratic Formula

Subtract 1.8x from both sides.

$$2x^{2}-1.8x=-1$$

Add 1 to both sides.

$$2x^{2}-1.8x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -1.8 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1.8\right)±\sqrt{\left(-1.8\right)^{2}-4\times 2}}{2\times 2}$$

Square -1.8 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-1.8\right)±\sqrt{3.24-4\times 2}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-1.8\right)±\sqrt{3.24-8}}{2\times 2}$$

Add 3.24 to -8.

$$x=\frac{-\left(-1.8\right)±\sqrt{-4.76}}{2\times 2}$$

Take the square root of -4.76.

$$x=\frac{-\left(-1.8\right)±\frac{\sqrt{119}i}{5}}{2\times 2}$$

The opposite of -1.8 is 1.8.

$$x=\frac{1.8±\frac{\sqrt{119}i}{5}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{1.8±\frac{\sqrt{119}i}{5}}{4}$$

Now solve the equation x=\frac{1.8±\frac{\sqrt{119}i}{5}}{4} when ± is plus. Add 1.8 to \frac{i\sqrt{119}}{5}.

$$x=\frac{9+\sqrt{119}i}{4\times 5}$$

Divide \frac{9+i\sqrt{119}}{5} by 4.

$$x=\frac{9+\sqrt{119}i}{20}$$

Now solve the equation x=\frac{1.8±\frac{\sqrt{119}i}{5}}{4} when ± is minus. Subtract \frac{i\sqrt{119}}{5} from 1.8.

$$x=\frac{-\sqrt{119}i+9}{4\times 5}$$

Divide \frac{9-i\sqrt{119}}{5} by 4.

$$x=\frac{-\sqrt{119}i+9}{20}$$

The equation is now solved.

$$x=\frac{9+\sqrt{119}i}{20} x=\frac{-\sqrt{119}i+9}{20}$$

Steps for Completing the Square

Subtract 1.8x from both sides.

$$2x^{2}-1.8x=-1$$

Divide both sides by 2.

$$\frac{2x^{2}-1.8x}{2}=-\frac{1}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{1.8}{2}\right)x=-\frac{1}{2}$$

Divide -1.8 by 2.

$$x^{2}-0.9x=-\frac{1}{2}$$

Divide -0.9, the coefficient of the x term, by 2 to get -0.45. Then add the square of -0.45 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-0.9x+\left(-0.45\right)^{2}=-\frac{1}{2}+\left(-0.45\right)^{2}$$

Square -0.45 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-0.9x+0.2025=-\frac{1}{2}+0.2025$$

Add -\frac{1}{2} to 0.2025 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-0.9x+0.2025=-\frac{119}{400}$$

Factor x^{2}-0.9x+0.2025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0.45\right)^{2}=-\frac{119}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-0.45\right)^{2}}=\sqrt{-\frac{119}{400}}$$

Simplify.

$$x-0.45=\frac{\sqrt{119}i}{20} x-0.45=-\frac{\sqrt{119}i}{20}$$

Add 0.45 to both sides of the equation.

$$x=\frac{9+\sqrt{119}i}{20} x=\frac{-\sqrt{119}i+9}{20}$$