Instead of using the quadratic formula to solve quadratics, we may be able to factor it instead. Factoring is an important part of simplifying and solving equations and will make the problem easier to solve. There are two main types of factoring that we will review: common factoring and factoring quadratics by decomposition.
Common Factoring
Common factoring is the type of factoring one should attempt first as it has the possibly of helping turn a tough problem into a relatively easy one. The idea behind common factoring comes from the distributive property of multiplication and finding the greatest common divisor (GCD) of the terms.
Below are the steps and an example to demonstrate common factoring.
Common factor 4x2 + 8x.
Step 1: Find the GCD of the individual terms.
The GCD of 4x2 and 8x is 4x.
Step 2: Divide the terms by the GCD.
4x2 + 8x divided by 4x is x + 2.
Step 3: Place the answer to step 2 in brackets and multiple it by the GCD.
4x(x + 2)
Common factoring is useful in finding roots for function as well as simplifying problems with large numbers.
Decomposition Method for Factoring Quadratics
For a quadratic ax2 + bx + c, we can attempt to factor it using the decomposition method. Below are the steps and an example to demonstrate the decomposition method.
Factor 6x2 + 31x + 35.
Step 1: Calculate the value of a*c for the quadratic ax2 + bx + c.
a*c = 6*35 = 210
Step 2: Find two numbers that when multiplied equal a*c, and when added equal b.
The numbers are 10 and 21, as we can see that 10 + 21 = 31 and 10 * 21 = 210.
Step 3: Write the middle term in the quadratic as the two numbers found.
6x2 + 31x + 35 = 6x2 + 21x + 10x + 35
Step 4: Common factor the terms in groups of two.
6x2 + 21x + 10x + 35 = 3x(2x + 7) + 5 (2x + 7)
Step 5: Common factor out the bracketed term.
3x(2x + 7) + 5 (2x + 7) = (2x + 7)(2x + 5)
There we know that 6x2 + 31x + 35 factors into (2x + 7)(2x + 5).
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This article was written for you by Jeremie, one of the tutors with SchoolTutoring Academy.
