Solve x^2-2x+32=0.3

$$x ^ { 2 } - 2 x + 32 = 0.3$$

Solve for x (complex solution)

$$x=\frac{i\sqrt{3070}}{10}+1\approx 1+5.540758071i$$

$$x=-\frac{i\sqrt{3070}}{10}+1\approx 1-5.540758071i$$

Steps Using the Quadratic Formula

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.

$$x^{2}-2x+32=0.3$$

Subtract 0.3 from both sides of the equation.

$$x^{2}-2x+32-0.3=0.3-0.3$$

Subtracting 0.3 from itself leaves 0.

$$x^{2}-2x+32-0.3=0$$

Subtract 0.3 from 32.

$$x^{2}-2x+31.7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and 31.7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 31.7}}{2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 31.7}}{2}$$

Multiply -4 times 31.7.

$$x=\frac{-\left(-2\right)±\sqrt{4-126.8}}{2}$$

Add 4 to -126.8.

$$x=\frac{-\left(-2\right)±\sqrt{-122.8}}{2}$$

Take the square root of -122.8.

$$x=\frac{-\left(-2\right)±\frac{\sqrt{3070}i}{5}}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±\frac{\sqrt{3070}i}{5}}{2}$$

Now solve the equation x=\frac{2±\frac{\sqrt{3070}i}{5}}{2} when ± is plus. Add 2 to \frac{i\sqrt{3070}}{5}.

$$x=\frac{\frac{\sqrt{3070}i}{5}+2}{2}$$

Divide 2+\frac{i\sqrt{3070}}{5} by 2.

$$x=\frac{\sqrt{3070}i}{10}+1$$

Now solve the equation x=\frac{2±\frac{\sqrt{3070}i}{5}}{2} when ± is minus. Subtract \frac{i\sqrt{3070}}{5} from 2.

$$x=\frac{-\frac{\sqrt{3070}i}{5}+2}{2}$$

Divide 2-\frac{i\sqrt{3070}}{5} by 2.

$$x=-\frac{\sqrt{3070}i}{10}+1$$

The equation is now solved.

$$x=\frac{\sqrt{3070}i}{10}+1 x=-\frac{\sqrt{3070}i}{10}+1$$

Steps for Completing the Square

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.

$$x^{2}-2x+32=0.3$$

Subtract 32 from both sides of the equation.

$$x^{2}-2x+32-32=0.3-32$$

Subtracting 32 from itself leaves 0.

$$x^{2}-2x=0.3-32$$

Subtract 32 from 0.3.

$$x^{2}-2x=-31.7$$

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.

$$x^{2}-2x+1=-31.7+1$$

Add -31.7 to 1.

$$x^{2}-2x+1=-30.7$$

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}.

$$\left(x-1\right)^{2}=-30.7$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-30.7}$$

Simplify.

$$x-1=\frac{\sqrt{3070}i}{10} x-1=-\frac{\sqrt{3070}i}{10}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{3070}i}{10}+1 x=-\frac{\sqrt{3070}i}{10}+1$$