Solving Quadratics and Complex Numbers

A quadratic equation is a polynomial equation of degree 2, typically in the form $$ax^2 + bx + c = 0$$ , where a, b, and c are constants and a ≠ 0.

The methods to solve quadratic equations include factoring, using the quadratic formula, completing the square, and taking square roots.

The quadratic formula is $$x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$$ , used to find the solutions of a quadratic equation.

To solve by factoring, express the quadratic in the form $$(x - p)(x - q) = 0$$ , then set each factor to zero: $$x - p = 0$$ and $$x - q = 0$$ .

The discriminant is given by $$D = b^2 - 4ac$$ . It determines the nature of the roots: if D > 0, two distinct real roots; D = 0, one real root; D < 0, two complex roots.

A complex number is a number of the form $$a + bi$$ , where a and b are real numbers, and i is the imaginary unit, defined as $$i = \sqrt{-1}$$ .

To add complex numbers, combine the real parts and the imaginary parts separately: if $$z_1 = a + bi$$ and $$z_2 = c + di$$ , then $$z_1 + z_2 = (a+c) + (b+d)i$$ .

To multiply complex numbers, use the distributive property: if $$z_1 = a + bi$$ and $$z_2 = c + di$$ , then $$z_1 z_2 = (ac - bd) + (ad + bc)i$$ .

Completing the square involves rewriting a quadratic equation in the form $$(x - p)^2 = q$$ to easily solve for x.

To simplify, combine the real parts and the imaginary parts: $$(a + c) + (b + d)i$$ .