Algebra Know-It-ALL

364 Quadratic Equations with Real Roots

As you know, all cases of subtraction are sums in disguise. The above expressions can be writ-

ten as pure sums like this:

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 )

Polynomial standard form

Any single-variable quadratic, no matter how complicated it looks when you first see it, can

be rewritten so it appears in polynomial standard form. When the equation is in this form, the

left side of the equals sign contains a constant multiple of the variable squared, added to a

constant multiple of the variable itself, added to a constant. On the right side of the equals

sign, you find 0 all by itself. Here’s the general equation:

ax^2 +bx+c= 0

where a,b, and c are constants, and x is the variable. All of the following are quadratic equa-

tions in polynomial standard form:

3 x^2 + 2 x+ 5 = 0

3 x^2 − 4 x= 0

− 7 x^2 − 5 = 0

− 4 x^2 = 0

In all but the first of these, some of the constants equal 0. But the first constant, the one by

whichx^2 is multiplied, can never be 0 in a quadratic equation. If you set a= 0 in the standard

form of a single-variable quadratic equation, you’ll get the standard form for single-variable

first-degree equation, because the term containing x^2 vanishes:

bx+c= 0

Mutant quadratics

Quadratic equations frequently appear in disguise. I call them mutant quadratics. Every such

equation has two things in common. First, it isn’t in polynomial standard form. Second, it can

be morphed into that form without changing the set of roots we get when we solve it. Here

are some examples:

x^2 = 2 x+ 3

x= 4 x^2 − 7

x^2 + 4 x= 7 +x

x− 2 =− 8 x^2 − 22

3 +x= 2 x^2