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Welcome to our Math lesson on Special Cases of Quadratic Equations, this is the second lesson of our suite of math lessons covering the topic of Quadratic Equations, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
Special Cases of Quadratic Equations that are solved by Factorizing
Although we did not suggest the use of the factorization method when solving quadratic equations, there are some cases in which this method is very suitable. More precisely, it is better to factorize when the equation has no constant (c = 0). In this case, it is not necessary to apply the other methods explained in the following paragraphs. The general form of such factorizations therefore is
ax2 + bx = x(ax + b) = 0
For example, in the quadratic equation
3x2 - 7x = 0
we can factorize x and as a result, we obtain
x(3x - 7) = 0
It is that simple! No extra steps, no challenging procedures. All you need to do is to identify all common factors in the equation terms and factorize them. The rest of the equation is written in brackets.
Thus, given that this equation is true for x = 0 or 3x - 7 = 0, its roots are x = 0 and x = 7/3.
Example 2
Find the roots of the following quadratic equation by factorization.
- 5x2 + 12x + 5 = 5
- 3 - 4x + 2x2 = 12 - 9
Solution 2
- First, we must turn both quadratic equations in the form ax2 + bx + c = 0. Thus, for the first equation, we have
5x2 + 12x + 5 = 5
5x2 + 12x + 5 - 5 = 5 - 5
5x2 + 12x = 0
Thus, factorizing x yields x(5x + 12) = 0
The left part is zero only when x = 0 or 5x + 12 = 0. Thus, the first root is x1 = 0. As for the second root, we have 5x + 12 = 0
5x = - 12
x = - 12/5
= - 2.4
Thus, x2 = 0. - Again, we must write the equation in the form ax2 + bx + c = 0. We have
3 - 4x + 2x2 = 12 - 9
3 - 4x + 2x2 = 3
- 4x + 2x2 = 3 - 3
- 4x + 2x2 = 0
2x2 - 4x = 0
This time we factorize 2x. Thus, we have 2x(x - 2) = 0
The left part of the last equation is zero, so must be the right part as well. Thus, this equation has two roots: the first is x1 = 0 because 2x = 0
x = 0/2
x = 0
The other root is calculated by finding the value of the variable in the part in brackets. Thus, we have x - 2 = 0
x = 2
Therefore, the second root of this equation is x2 = 2.
More Quadratic Equations Lessons and Learning Resources
Equations Learning Material| Tutorial ID | Math Tutorial Title | Tutorial | Video
Tutorial | Revision
Notes | Revision
Questions |
|---|
| 9.5 | Quadratic Equations | | | | |
| Lesson ID | Math Lesson Title | Lesson | Video
Lesson |
|---|
| 9.5.1 | Solving Quadratic Equations by Factorizing | | |
| 9.5.2 | Special Cases of Quadratic Equations | | |
| 9.5.3 | Solving Quadratic Equations by Factorization Part Two | | |
| 9.5.4 | Solving a Quadratic Equation by Completing the Square | | |
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- Continuing learning equations - read our next math tutorial: The Quadratic Formula
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