which reduces to 7x^2 /2. This allows us to rewrite the right side of the original equation to obtain
7 x^2 /2+ 7 x− 5 = 7 x^2 /2+ 3 x
Subtracting 7x^2 /2 from each side, we get
7 x− 5 = 3 x
When we subtract 3x from each side now, we get an equation in the standard single-variable, first-degree
form:
4 x− 5 = 0
Binomial Factor Form
There’s another way to express a quadratic equation: as a product of two binomials that is
equal to 0. In some ways, this form is simpler than the polynomial standard form. As you’ll
soon see, the binomial factor form of a quadratic tells you the roots directly.
Binomials in quadratics
When a left side of a quadratic is expressed as a product of binomials, both of the binomi-
als must be in a specific form. The first term in each binomial is a multiple of the variable.
The multiplicand in that term is the coefficient of the variable. The second term is a constant,
sometimes called the stand-alone constant. Here are some examples:
x+ 1
x− 5
3 x+ 5
− 17 x+ 24
8 x− 13
− 7 x− 11
Multiplying two binomials
If we multiply two binomials of the above sort where both binomials contain the same vari-
able, and if we then set the product equal to 0, we get a quadratic equation. Here’s the general
binomial factor form for a quadratic:
(px+q)(rx+s)= 0
where p and r are the coefficients, q and s are the constants, and x is the variable.
To produce a quadratic, neither of the coefficients p nor r can be equal to 0. Other than
that, there’s no restriction on the values of the coefficients and constants. As long as p, q, r, and
s are all real numbers, then the resulting equation will have real roots.
Binomial Factor Form 367
