Factorise 5√5x^2+30x+8√5 by spliting the middle term

Question

This is from the chapter polynomial.

Answer 1

5sqrt5*x^2 + 30x + 8sqrt5 = 0

A = 5sqrt5 , B = 30 , and C = 8sqrt5

          -B +/- sqrt(B^2 - 4AC)        -30+/_ sqrt( 30^2 - 4*5sqrt5*8sqrt5)       -30 +/- sqrt(900-800)

X = --------------------------------- = -------------------------------------------------= ---------------------------------------

                      2A                                                2*5sqrt5                                     10sqrt5

x = (-30- 10)/(10sqrt5)= -4/sqrt5 = -4sqrt5/5

x = (-3-+10)/(10sqrt5) = -2sqrt5/5

5sqrt5*x^2 +30x + 8sqrt5 = (x + 4sqrt5/5)(x +2sqrt5/5)

Answer 2

We can express the factors as (5x√5+a)(x+b) where ab=8√5, so b=8√5/a, and 5b√5+a=30, the coefficient of the middle term. Therefore (5√5)(8√5/a)+a=30, that is, 200/a+a=30; 200+a^2=30a and a^2-30a+200=0=(a-20)(a-10).

Therefore, b=8√5/20=2√5/5 or b=4√5/5 and the factors are:

(5x√5+20)(x+2√5/5) or (5x√5+10)(x+4√5/5).

These factorisations are equivalent: the middle term, 30x, is split into (5√5)(2√5/5)x+20x=10x+20x, or (5√5)(4√5/5)x+10x=20x+10x.