Quadratic Equations and Inequalities

Complex Numbers

The number \(i\) is not a real number and is often called the imaginary unit, but the number \(i^2\) is the real number \(-1\).

The form \(a + bi\) is called the standard form of a complex number. The real number \(a\) is called the real part of the complex number, and \(b\) is called the imaginary part.

Two complex numbers \(a + bi\) and \(c + di\) are said to be equal if and only if \(a = c\) and \(b = d\).

Adding and Subtracting Complex Numbers

To add complex numbers, we simply add their real parts and add their imaginary parts.

$$ (a + bi) + (c + di) = (a + c) + (b + d)i $$

The set of complex numbers is closed with respect to addition; that is, the sum of two complex numbers is a complex number. Furthermore, the commutative and associative properties of addition hold for all complex numbers.

Simplifying Radicals Involving Negative Numbers

In the set of complex numbers every negative real number has two square roots because:

$$ (i \sqrt{b})(i \sqrt{b}) = i^2b = -1b $$

so we see that:

$$ \sqrt{-b} = i\sqrt{b} $$

Multiplying Complex Numbers

“e say that two complex numbers \(a + bi\) and \(a - bi\) are called conjugates of each other. The product of a complex number and its conjugate is always a real number.

$$ (a + bi)(a - bi) = a(a - bi) + bi(a - bi) \\ = a^2 - abi + abi - b^2i^2 \\ = a^2 - b^2(-1) \\ = a^2 + b^2 $$

Dividing Complex Numbers

We use conjugates to simplify expressions such as \(\frac{3i}{5 + 2i}\) that indicate the quotient of two complex numbers.

Quadratic Equations

A quadratic equation in the variable x can also be defined as any equation that can be written in the form:

$$ ax^2 + bx + c = 0 $$

where \(a\), \(b\) and \(c\) are real numbers and \(a \neq 0\). The previous expression is referred as the standard form.

Solving Quadratic Equations of the Form \(x^2 = a\)

$$ x^2 = a \\ x^2 - a = 0 \\ x^2 - (\sqrt{a})^2 = 0 \ (x - \sqrt{a})(x + \sqrt{a}) = 0 $$

Therefore

$$ x - \sqrt{a} = 0 \\ x = \sqrt{a} $$

or

$$ x + \sqrt{a} = 0 x = -\sqrt{a} $$

Completing the Square

In this section we examine another method called completing the square, which will give us the power to solve any quadratic equation.

For example to solve \(x^2 + 10x - 2 = 0\):

$$ x^2 + 10x - 2 = 0 \\ x^2 + 10x = 2 \\ x^2 + 2(\frac{10}{2})x = 2 \\ x^2 + 2(5)x = 2 \\ x^2 + 10x + 5^2 = 2 + 5^2 \\ (x + 5)^2 = 27 \\ x + 5 = \pm \sqrt{27} \\ x + 5 = 3 \pm \sqrt{3} \\ x = -5 + 3 \pm \sqrt{3} \\ $$

Quadratic Formula

Determining the Nature of Roots of Quadratic Equations

The quadratic formula makes it easy to determine the nature of the roots of a quadratic equation without completely solving the equation. The number

$$ b^2 - 4ac $$

which appears under the radical sign in the quadratic formula, is called the discriminant of the quadratic equation. The discriminant is the indicator of the kind of roots the equation has.

1. If \(b^2 - 4ac < 0\), then the equation has two nonreal complex solutions.
2. If \(b^2 - 4ac = 0\), then the equation has one real solution.
3. If \(b^2 - 4ac > 0\), then the equation has two real solutions.

We make the statement that if \(b^2 - 4ac = 0 \), then the equation has one real solution. Technically, such an equation has two solutions, but they are equal. We sometimes refer to this as one real solution with a multiplicity of two.

We refer to each of the solutions as critical numbers.