This article is going to give you a detailed explanation of quadratic equation, its meaning, different methods of solving them using factorization method. Ensure that you read this article to the end in order to be fully acquainted with the easy steps for solving quadratic equation.
At the end of this article, you will be able to solve simple and complex problems involving simultaneous equation using the factorization method. You will also know when to use the factorization method and when to use other methods of solving simultaneous equations.
What Is Quadratic Equation?
Quadratic equation is one which in the highest power of the unknown is 2. This indicates that you will have two values of the unknown that will satisfy the equation when you check. The values of the unknown are sometimes called the roots of the equation.
Quadratic equation takes the general formula of:
ax2 + bx + c = 0
a, b and c are constant such that a≠0. If a = 0, then it is not a quadratic equation.
Examples Of Quadratic Equations
The following are some examples of a quadratic equation:
X2 + 7x + 10 = 0
X2 – x – 20 = 0
4x2 = 9
(x-1)(x-2) = 0
(x+2)2 = 25
In all the above examples, you can see that all of them follow the general quadratic equation formula of ax2 + bx + c = 0. What actually makes an equation to be seen as a quadratic is the value of the highest power of the unknown (x in our case) which should not be greater than 2.
How Do I Solve Quadratic Equations?
There are four (4) ways to solve a quadratic equation. They include the following:
- Factorization method
- Completing the squares
- Formula Method
- Graphical Method
Each of the above methods can solve a wide range of quadratic equation. In this article, however, we are going to focus on solving quadratic equation using factorization method. We will try to solve different exercises to ensure your mastery of the topic and also we are going to highlight the limitations to the use of factorization method.
Factorization Method Of Solving Quadratic Equation
Factorization involves simplifying an equation using common factors of the equation. When we consider the general quadratic equation of ax2 + bx + c = 0, factorization method will involve expressing this given equation as two linear factors (x + m) and (x + n) in such a way that ax2 + bx + c = (x + m)(x + n) = 0. These factors can further be explained as:
If (x + m)(x + n) = 0
Then; x + m =0……………………………….(1)
x + n = 0………………………………..(2)
The solutions of these two linear equations give the solution of the quadratic equation as they satisfy the values of the unknowns.
Solve the quadratic equation x2 – 4x -12 = 0 using the factorization method
x2 – 4x – 12 = 0
STEP 1: Find two common factors which when simplified give the quadratic equation in question. You can do this easily by multiplying the integer with the unknown that has power. That is x2 multiplied by -12. List all the common multiples of -12 and select one in which you will get the middle term (-4x) if you simplify. The two multiples are +2 and -6.
Therefore, in the equation above, the factors are (x + 2) and (x – 6).
STEP 2: Express the factors in an equation form
(x + 2)(x -6) = 0
Note that, when you expand this equation, you will get the quadratic equation x2 – 4x – 12 = 0, which means that the factors are correct.
From the equation, (x + 2)(x -6) = 0
Either x +2 = 0
Or x – 6 = 0
Therefore, x= -2 or +6
Solve the quadratic equation (x -4)(x +5)=0
The expression (x -4)(x +5) has been factorized already, so you can go ahead to simplify to get the two values of the unknown.
(x -4)(x +5) = 0
Either x -4 = 0
x = 4
or x + 5 = 0
x = -5
Therefore, x = 4 or -5
Solve the given quadratic equation (x + 2)2= 0 using Factorization method
There are two ways to solve this; one of them is very easy and straightforward. We shall proceed with the longer way before moving ahead with the easy method.
(x + 2)2= 0
Step 1= Open Bracket
(x + 2)2= 0
X2 + 4x + 4 = 0
Step 2; Get the two common factors of the equation
The two factors are +2 and +2
(x + 2)(x+2) =0
Either x + 2 = 0
X = -2
0r x + 2 = 0
x = -2
Therefore, the two values of the unknown are -2 and -2.
Alternatively, you can solve it this way
(x + 2)2= 0
This can be written as;
(x + 2)(x + 2) =0
You can then go ahead and complete the solving.
Using Factorization method, solve the quadratic equation 2x2 + 6x = 0
You have to find the two factors of the equation
2x2 + 6x = 0
2x(x+3) = 0
Either 2x = 0
x = 0/2
Or x + 3 = 0
x = -3
Therefore x = 0 or -3
Solve the equation, x2 = 9
x2 = 9
Move all the numbers to the left hand side of the equation leaving only zero at the right hand side before factorization
x2 – 9 = 0
Factorize using difference of two squares
Recall, x2 – 9 = x2 – 32
x2 – 32 = 0
(x-3)(x+3) = 0
Either x – 3 = 0
x = 3
or x + 3 = 0
x = -3
Therefore, x = 3 or -3
Limitations Of Factorization Method of Solving Quadratic Equation
The factorization method has some significant limitations:
- It cannot solve all quadratic equations as not all equations can be factorized.
- When the coefficient of the unknown have many factors, it becomes more difficult to list out all the factor pairs that you need to solve the equation.
NOTE: Always be mindful of the negative signs when you are carrying out factorization.
This article has been able to clearly demonstrate how to solve a quadratic equation using the factorization method. If you have any questions regarding this, please use the comment section below.
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