Algebra Know-It-ALL

363

CHAPTER

22 Quadratic Equations with Real Roots

In this chapter, you’ll learn about single-variable quadratic equations, often called quadratics for

short. First, you’ll see how you can recognize one. Then you’ll learn how you can solve it if there’s at

least one solution, called a root, in the set of real numbers. Sometimes a quadratic equation has two

real roots, sometimes it has only one real root, and sometimes it has no real roots.

Second-Degree Polynomials

All quadratics are second-degree equations. That means they can be written in a form that con-

tains a variable raised to the second power (squared), but no power higher than that.

Polynomials

A quadratic equation can always be portrayed as a polynomial, which means “expression with

multiple names,” on the left side of the equals sign, with 0 on the right side. Here are some

examples of polynomials. Only the second of these can form a quadratic if we set it equal to 0.

x− 3

−x^2 + 3 x− 6

x−y^3 + 7 z^2

2 x− 2 y^5 − 2 z^7

ax^4 y−bxz^3 −cy^2 z^2

where a,b, and c are constants, and x,y, and z are variables. Polynomials contain parts called

terms that are added together. The terms in a polynomial are also known as monomials. Some-

times the monomials are complicated; they can even be polynomials themselves. A polynomial

can contain any finite number of monomials—hundreds, thousands, or millions perhaps—

but you’ll rarely see a polynomial with more than a dozen terms.

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