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One of the important methods for solving quadratic equations is factoring. The great benefit of this method lies in shortening the time it takes to apply the traditional method using the general quadratic formula. There are equations that can be easily factored and their roots found even by simply looking at the equation.
Instead of applying the general formula and the delta root, and wasting time, we can easily find the roots through factoring skills.
Methods for Solving a Quadratic Equation
There are many ways to solve a quadratic equation, including:
- Solving a quadratic equation using the general formula
- Solving a quadratic equation by completing the square
- Solving a quadratic equation by factoring.
The third method is what we will discuss in detail today in our article. However, we can give you some tips for choosing the most suitable method for you to solve:
- If the equation is easily factorable, then factoring is the fastest method.
- The general formula (delta) method is a guaranteed way to solve any quadratic equation (even if the equation has no solution in the set of real numbers).
- The completing the square method is useful for understanding the general formula but is less common for direct solutions.
As a quick review, let’s recall the general form of a quadratic equation before we begin explaining the method.
The Quadratic Equation
A quadratic equation is a mathematical equation whose basic general form is as follows:
ax² + bx + c = 0
Where:
- a, b, c are constants (numerical values).
- x is the variable whose value we want to find (the roots or solutions of the equation).
The condition for the equation is that a is not equal to zero. If a were equal to zero, it would mean the term containing x2 vanishes, and thus the equation would reduce to a linear equation.
There are several ways to solve a quadratic equation, including the general method we discussed in the article (Quadratic Equation Solver) and the factoring method, which we will talk about in today’s article.
Solving a Quadratic Equation by Factoring
Before explaining the method, it is important to note that this method cannot always be applied to find the factors. Therefore, it is an experimental method for saving time. When factoring is not possible, it is necessary to resort to the traditional method using the general formula.
Always ensure the solutions are correct when using the factoring method, as finding factors is done through mental calculation. Verification is done by substituting the solutions back into the equation to confirm they satisfy it.
The idea behind factoring relies on writing the equation in the form of a product of two binomials:
(x + d) (x + e) = 0
Time Required: 2 minutes.
Solve a quadratic equation by factoring
- Write the Equation in General Form We must ensure that the equation is written in general form and simplified appropriately. The general form is:
ax² + bx + c = 0 - Case One: a = 1 Look for two numbers whose product is = c and whose sum is = b (only if a = 1).
- Case Two: a is not equal to 1 The factoring process is more complex and requires trial and error or other methods.
- Find the Factors After Factoring If you find the two numbers (d) and (e), you can set each parenthesis equal to zero. The solutions to the equation will be:
Either x + d = 0 => x = -d
Or x + e = 0 => x = -e
Since if there are two parentheses multiplied together in an equation equal to zero, at least one of them must be equal to zero.
Examples of Solving a Quadratic Equation by Factoring
Solve the following equation by factoring:
x² + 5x + 6 = 0
Solution: We are looking for two numbers whose product is 6 and whose sum is 5. The numbers are simply 2 and 3.
Now, write the equation as a product of two binomials (factors):
(x + 2) (x + 3) = 0
Thus, the solutions to the equation are:
Either x + 2 = 0 => x = -2
Or x + 3 = 0 => x = -3
So the solutions are [-3, -2]
Exercise 2: Find the solution to the following equation by factoring:
x² – 8x + 15 = 0
Solution: We are looking for two numbers whose product equals the constant term (15) and whose sum equals the coefficient of the middle term (-8).
The numbers are -3 and -5, because (-3) × (-5) = 15 and (-3) + (-5) = -8. (We can say directly that the solutions are 3 and 5, which are the opposite signs of the factors we found. However, in an exam, the full method should be written).
Write the equation in factored form using these two numbers:
(x – 3) (x – 5) = 0
Either x – 3 = 0 => x = 3
Or x – 5 = 0 => x = 5
Thus, the values of x that satisfy the equation are 3 and 5.
Exercise 3: Solve the following equation by factoring:
2x² + 10x = 0
We notice a common factor between the two terms, which is 2x. This means we can factor out 2x as follows:
2x (x + 5) = 0
Now we have the equation in the form of a product of two terms. We can say:
Either 2x = 0 … x = 0
Or x + 5 = 0 … x = -5
Thus, the solutions to the equation are 0 and -5
Conclusion
Not all quadratic equations are factorable. In such cases, we resort to other methods like the general quadratic formula or completing the square. It is also very important to verify the solutions we find by substituting them back into the equation.
Generally, we resort to the factoring method in exams when we are not required to solve the equation directly but rather perform other tasks. In these situations, we use this method for speed only.
