Question

What is the shortcut for middle term splitting ?

Answer 1

I will explain this with the help of an example

Suppose x²+3x-10
To split this the term 3x should be splitted in such a way that the splotted terms when multipled with each other the result is equal to the product of the first and last terms of the quadratic equation and the sum of the splitted terms is the middle term of the quadratic equation.

To split this equation x²+3x-10,
first we multiply the first and last terms

That is x²×-10=-10x<sup>2

Now the factor of this should be taken</sup>
10 ⇒ 1×10
2×5
Now from the above we can find the correct one to split by the condition that on multiplying both we should get -10 and on adding +3

As the result of addition is positive the greater number should be positive and the smaller number should be negative to satisfy the above condition that the product should be -10 which is negative and the sum should be positive 3

We can take both the above factor multiplication

1)10-1=9
or
10x-x=9x
The sum is not 3 or 3x. So this cannot be taken

2)5-2=3
or
5x-2x=3x
The sum is 3x and also its product is 5x×-2x=-10x
Therefore this can be used

∴ x²+3x-10=x²+5x-2x-10
=x(x+5)-2(x+5)
=(x+5)(x-2)
∴x+5=0 and x-2=0 are the solutions
i.

That is x= -5 or x=2

Similarly,take another example

x²+7x+12

12× x²=12x²

12 ⇒ 1×12
2×6
3×4
1+12≠7
2+6≠7
3+4=7

∴ 7x can be splitted as 3x+4x
x²+7x+12= x²+3x+4x+12
=x(x+3)+4(x+3)
=(x+3)(x+4)
Therefore x=-3 and x=-4