Factorization: Quadratics
Factorization of quadratics involves breaking down quadratic expressions, typically in the form \( ax^2 + bx + c \), into simpler linear factors. This process is grounded in the Fundamental Theorem of Algebra, which states that any quadratic expression can be expressed as the product of two linear factors. There are several methods available for factorizing quadratics, including the difference of two squares, completing the square, the quadratic formula, and inspection.
Each method has its own applications and effectiveness: the first three methods consistently yield results, while inspection is more suitable for simpler quadratics and requires more skill. The discriminant, a component derived from the coefficients \( a \), \( b \), and \( c \), provides valuable insight into the nature of the quadratic's roots—whether they are real and distinct, repeated, or complex. Understanding these methodologies and the role of the discriminant allows for a comprehensive approach to solving quadratic equations, making factorization an essential skill in algebra.
Factorization: Quadratics
A quadratic expression is of the form
where a, b, and c can be either real or complex constants.
By the Fundamental Theorem of Algebra, a quadratic expression can always be factored into two linear factors, that is ax2 + bx + c = a(x + p)(x + q) where p and q might be either real or complex constants.
There are four commonly used methods to factor a quadratic: 1) difference of two squares, 2) completing the square, 3) the quadratic formula, and 4) inspection. The first three always work. Inspection works only for the simplest quadratics and requires some skill.
Discriminant
The expression
is called the "discriminant" of the quadratic expression ax2 + bx + c. If a, b, and c are real numbers, then the discriminant tells what kind of factors the quadratic has.
b2 – 4ac > 0 the quadratic has two distinct real factors.
b2 – 4ac = 0 the quadratic has one real factor, repeated twice.
b2 – 4ac < 0 the quadratic has two complex factors which are conjugates of each other, that is, the factors are (px + q) and (px – q) where q is an imaginary number.
If one or more of a, b, and c is complex, the factorization is likely to involve complex quantities.
Difference of Two Squares
This method can only be used for ax2 + bx + c when b = 0. Then ax2 − c factors into
(x√a +√c)(x√a −√c) and ax2 + c factors into (x√a +i√c)(x√a −i√c) where i = √(−i).
Completing the Square
The idea behind completing the square is to render the quadratic ax2 + bx + c into the form a(x2 − p2) (which is called difference of two squares) which factors into a(x + p)( x − p).
First, if a ≠ 1, factor out a to get a(x2 + (b/a)x + c/a).
Now add and then subtract the quantity b2/4a2. This quantity is one-half of b/a, squared. The result is
Note that the expression in [brackets] is the square of (x + b/2a). Note that 4a2 is the square of 2a. The desired factorization, by the difference of two squares formula, is
Quadratic Formula
The quadratic formula
is derived from formula (4). To use the quadratic formula to factor ax2 + bx + c, simply insert a, b, and c into (4).
Inspection
This is the usual first method taught to beginning algebra students. In the simplest case, consider a quadratic x2 + bx + c where b and c are integers. If (5) can be factored into (x + p)(x + q), then the factors multiply to (x2 + (p+q)x + pq). One must find p and q such that p + q = b and pq = c.
For example, consider x2 + 7x + 10. In this case p + q = 7 and pq = 10 and the only integers p and q for which this is true are 2 and 5. Hence x2 + 7x + 10 = (x + 2)(x + 5).
For a more difficult example, consider x2 + 4x – 12. It is necessary to find 2 integers such that their sum is 4 and their product is −12. −1 and 12 give a sum −11, so they are not correct. Neither is 1 and −12. 2 and −6 give a sum of −4, so they are not correct. However, −2 and 6 give a sum of 4, so x2 + 4x – 12 = (x – 2)(x + 6) is the desired factorization.
The inspection method can be applied to more complicated quadratics, but this requires some skill, and at some point it is easier to use completing the square or the quadratic formula instead.
Bibliography
Francisco, Christopher. Closures, Finiteness and Factorization. Berlin: De Gruyter, 2012.
Larson, Ron, and David C. Falvo. Precalculus. Boston: Cengage, 2014.
Lial, Margaret L., E. J. Hornsby, and Terry McGinnis. Intermediate Algebra. Boston: Pearson, 2012.
Miller, Julie. College Algebra Essentials. New York: McGraw, 2013.
Stewart, James, L. Redlin, and Saleem Watson. Precalculus: Mathematics for Calculus. Belmont, CA: Cengage, 2012.
Young, Cynthia Y. College Algebra. Hoboken, NJ: Wiley, 2012.