Solving Quadratic Equations ┃Middle Term Splitting

 

Middle Term Splitting


Apart from plugging in the coefficients in the formula which solves quadratic equations, there are simpler ways of solving quadratic equations, without the use of calculators! The method described here is known as mid-term splitting. 

Benefits:

This method requires much less calculation and effort. It is solved in a quicker and cleaner method.

Drawbacks: 

Unlike the quadratic formula, the middle term splitting method only works in certain cases. 


How does it work:

ax² + bx + c

This method will only work if there are two numbers that add up to 'b' and have a product of 'a*c'


What we do is we write the equation in another form and factorise it into two parts which equal zero. The examples below will explain better!



Here, 6 and -5 are the two numbers needed. They are the set of numbers which add up to the origibal 'b' whoch was 1, and form a product of -30( a*c)! After that the equation was just factorised. Since this would equal to zero, we would get two sub equations:

x-5=0 and x+6=0

From here, we know the answers of the quadratic equation are 5 and -6!



In this case we need a set of numbers that multiply to 24 and add up to 5, thus, 8 and -3! Now we follow the same method as before and get the solutions/ roots as 3/2 and -4!



Questions                                                                                             


x2 -8x -9



Solution:

The middle term needs to be split into two terms whose product is -9x2 while the sum remains -8x
 x2 - 9x  + 1x  -9
= x(x - 9) + 1 (x - 9)
= (x - 9) (x + 1)




x2 +13x -168

Solution:

The middle term needs to be split into two terms whose product is -168x2 while the sum remains +13x
 x2 + 21x  - 8x  -168
= x(x + 21) - 8 (x + 21)
= (x + 21) (x - 8)




x2 +3x -40

Solution:

The middle term needs to be split into two terms whose product is -40x2 while the sum remains +3x
 x2 + 8x  - 5x  -40
= x(x + 8) - 5 (x + 8)
= (x + 8) (x - 5)



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