Factorizing quadratic equations is fairly tricky, and needs practice, but there are some easy-to-remember guidelines that can help.
Factorizing quadratic functions of equations is important because it is an important step in finding the roots or zeros of the equations. Factorizing causes problems for some students, even competent math students. The reason is that there are often many options to consider. Some tips are listed here to help factorise quickly.
Factorizing Quadratic Equations: Common Constant Factor
When each term in the equation contains a common factor, the factor may be removed. For example, 2 is a common factor in the equation
2.x² + 6.x + 4 = 2.(x² + 3.x + 2)
Factorizing Quadratic Equations: a²x² - b² and (ax + b)²
In this case, there is no "x" term, just terms in x² and a constant term e.g.
4.x² - 9
This is very easy to resolve, as it is the "difference of two squares". The solution is simply
(2.x - 3).(2.x + 3)
The equation may still be factorized even when a and b are not perfect squares:
3.x² - 2 can be factorized into
(√3. x - √2).(√3. x + √2)
A similar version of this is the "sum squared". The identity
(a.x + b)² = a².x² + 2.a.b.x. + b² ... allows a short-cut in some cases. For example,
(x + 1)² = x² + 2.x + 1
In general, if the square root of the constant term (1 in this case) is equal to half of the x-coefficient, then the expression is one factor squared. Another example of this is
(x + 3)² = x² + 6.x + 9
Factorizing Quadratic Equations: Coefficient Signs
The four equations following demonstrate how the in-bracket factorized signs define the signs of the un-factorized equations:
(x - 1).(x - 2) = x² - 3.x + 2 ...... equation [1]
(x + 1).(x + 2) = x² +3.x + 2 .... equation [2]
(x - 1).(x + 2) = x² + x - 2 ........ equation [3]
(x + 1).(x - 2) = x² - x - 2 ......... equation [4]
These four equations show all of the possible combinations for (x ± 1) × (x ± 2). Note that when the coefficient of the "x" term is negative, but the constant term is positive, then the factors are of the form (x - something), and when the "x" term and the constant term are both positive, then the factors are both of the form (x + something). Equations 3 and 4 are revealing, however. In these cases, the constant term is negative, but the term in "x" can be either positive or negative. More importantly, the sign of the term in "x" is the sign of the larger of the two factors. i.e. 2 is the largest factor of the constant term, so when its sign is positive (equation 3), then the term in "x" is also positive. Conversely, when the sign of 2 is negative (equation 4), then the sign of the term in "x" is also negative.
Factorizing Quadratic Equations: Choosing Factors of the Constant Term
The three equations following demonstrate how the factors of the constant term affect the coefficient of "x".
(x - 1).(x - 16) = x² - 17.x + 16 ... equation [5]
(x - 2).(x - 8) = x² -10.x + 16 ...... equation [6]
(x - 4).(x - 4) = x² - 8x + 16 ......... equation [7]
All three of these has an x² term and a constant term of +16. Furthermore, the coefficient of the x-term increases in size as the factors become further apart. This is very useful to bear in mind, because one of the main problems that students have with factorizing, is to choose the factors for the constant term. It is therefore useful to test the furthest apart factors (1 and 16 in this case), then the two closest together (4 and 4), to get a feel for which pair of factors of the constant term gives the desired result.
Quadratic Factorization: More Tips
After the suggestions above have been followed, the student with have a smaller range of possibilities from which to choose. One further tip is to check the coefficients of x² and x, and the constant factor. By looking at which of these are odd, and which are even, the range of factors may be reduced even more.
If all other options have been exhausted, then the general solution to quadratic equations may be used.
Quadratic Factorization Summary
There is a series of steps that may be used to reduce the number of possible factors, and hence the amount of confusion, for quadratic equations. These are listed here, along with a reference to the general quadratic solution formula. The tips used here may also be used to solve quadratic functions of other functions, such as trigonometric identities and others.
Factorising Quadratics Shouldn't Need a Calculator - image credit
Factorizing Quadratic Equations - References
Since the equations used here have been taken from basic mathematical identities, the origins of which in antiquity, no references are necessary.
Knowing Vieta's formulas and how it works will certainly help in understanding this process :)
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