A quadratic equation is also known as a trinomial. It has three terms in it: One is the term with `x^2`, the other is the term with `x` in it, and the third term is a constant number.
There are three parts to the method of splitting the middle term:
You need to first check if the given quadratic equation is in the standard form. A quadratic equation in the standard form looks like this:
Convert this quadratic equation into standard form:
Therefore the equation in standard form is:
The middle term of a quadratic equation is the term that has an 'x' in it but not the one that has 'x ^{2}' in it. For example, the middle term of the following quadratic equation
Now to split the middle term, you have to find two numbers (negative or positive) that add up to give the number in the middle term. For example, the following pairs of numbers add up to give the number -4 of the middle term:
Since -4x = -6x + 2x,
we can write -6x + 2x in place of -4x in the quadratic equation. That is,
After splitting the middle term of a quadratic equation, it becomes a four termed polynomial. That is, now it
In the equation
So take (x - 3) as the single common factor for both the groups and take it out as follows:
Interestingly, the equation you have obtained above by factorizing the quadratic equation is said to be in the intercept form.
There are three parts to the method of splitting the middle term:
- Rearrange the given quadratic equation into the standard form (if not already in the standard form)
- Split the middle term
- Factor the quadratic equation.
Rearrange the given quadratic equation into the standard form (if not already in the standard form):
You need to first check if the given quadratic equation is in the standard form. A quadratic equation in the standard form looks like this:
`ax^2 + bx + c = 0`For example, the following quadratic equations are in the standard form:
`2x^2 + 3x + 6 = 0`, `x^2 - 4x - 5 = 0`, and - `3x^2 - 6x + 5 = 0`Now if a quadratic equation is not given in the standard form, then we will write it in the standard form. Like this:
Convert this quadratic equation into standard form:
`2x^2 = 4x + 6`The steps will be like this:
Step | Explanation |
---|---|
2x ^{2} - 4x = 6 | Subtracted 4x from both sides |
2x ^{2} - 4x - 6 = 0 | Subtracted 6 from both sides |
Therefore the equation in standard form is:
`2x^2 - 4x - 6 = 0`Similarly you will rewrite all quadratic equations in to the standard form in order to split the middle term and factorize.
Split the middle term:
The middle term of a quadratic equation is the term that has an 'x' in it but not the one that has 'x ^{2}' in it. For example, the middle term of the following quadratic equation
`2x^2 - 4x - 6= 0`is `-4x`. Remember to take the sign at the left of the middle term as well.
Now to split the middle term, you have to find two numbers (negative or positive) that add up to give the number in the middle term. For example, the following pairs of numbers add up to give the number -4 of the middle term:
- -2 + -2 = -4
- -3 + -1 = -4
- -5 + 1 = -4
- -6 + 2 = -4
The product of the two parts of the middle term should be equal to the product of the first and last terms in the quadratic equation/expression.The first and last terms in the quadratic equation
`2x^2 - 4x - 6 = 0`are `2x^2` and `-6`. Their product is
`2x^2 * -6 = -12x^2`So if you split the number of the middle term into two parts ' a' and ' b', then their product should be -12. Which of the pairs of numbers mentioned above multiply to give -12?
-6 * 2 = -12There fore these two numbers -6 and 2 will form the correct two parts of the middle term -4x of the quadratic equation 2x ^{2} - 4x - 6 = 0 .
Since -4x = -6x + 2x,
we can write -6x + 2x in place of -4x in the quadratic equation. That is,
2x ^{2} - 4x - 6 = 0is same as
2x ^{2} - 6x + 2x - 6 = 0Similarly you will split the middle term of all other quadratic equations.
Factorize:
After splitting the middle term of a quadratic equation, it becomes a four termed polynomial. That is, now it
In the equation
2x ^{2} - 6x + 2x - 6 = 0Making groups of the first two and the last two terms,
(2x ^{2} - 6x) + (2x - 6) = 0Now take a common factor from each group: from the first group we take 2x as the common factor and from the second group we take 2 as the common factor:
2x(x - 3) + 2(x - 3) = 0Now you will always notice that there is a common factor in both groups. Here it is (x - 3) (Notice that (x - 3) is common in both the groups).
So take (x - 3) as the single common factor for both the groups and take it out as follows:
(x - 3)(2x + 2) = 0Now, to completely factorize the equation, you can take 2 as the common factor from the second parenthesis,
2(x - 3)(x + 1) = 0Therefore we factorized the quadratic equation 2x ^{2} - 4x - 6 = 0 into (x - 3)(2x + 2) by the method of splitting the middle term
Interestingly, the equation you have obtained above by factorizing the quadratic equation is said to be in the intercept form.
the only word i want say thanks you!!!!!
ReplyDeleteyea dis doesnt explain nethn y don u give more examples...
ReplyDelete@ Additional solved examples for splitting the middle term are in the post http://blogformathematics.blogspot.com/2011/09/additional-solved-examples-for_21.html
Deletehow to split large terms??
ReplyDeleteThere is no formula or approach for large numbers that can make it easier. First find the product of `a` and `c` in the equation `ax^2 + bx + c` and then prime factorize the product. Now using these factors try to find two numbers whose product is same as `ac` and sum is equal to `b`.
DeleteFor example `50x^2 +111x+ 22` can be factored as follows:
Product of first and third numbers: `50 * 22 = 1100`
Find two numbers such that their product is `1100` and sum is `111`, you get `11` and `100` and the equation becomes `50x^2 + 11x + 100x + 22`
HOW TO SPLIT IF THE TERMS A VERY LARGE LIKE SOME 3460??
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ReplyDeleteWhy do we split the middle why not the fist term
ReplyDeleteWe can split the first term, but it will not help us factor the expression. Splitting the middle term is about factoring the quadratic expression.
ReplyDeleteThe first term of a quadratic expression is the `x^2` term, while the middle term is the `x` term. If you want to factor the expression into a product of two linear expressions, splitting the `x^2` term does not make any sense because then you would not be able to take a common factor out of the four terms you get.