Quadratic equation: Part-3

5) Relation between roots and coefficients of Quadratic equation:
Let α and β be two roots of quadratic equation then:
α + β= -b/a (α .β)= c/a
Question : If α and β are roots of x2 +5x -1 = 0 find α3 + β3
solution: As by comparing with Standard form a= 1 b= 5 c= -1
α + β =-5 α .β =-1
As α3 + β3 = (α + β)3 -3 α.β( α + β)
= (-5)3 -3 (-1)(-5)
= -125 -15
=-140

Application Based Questions :
1) A train travels 360 km at a uniform speed. If the speed had been 5 km/hr more, it would have 1 hour less for same journey. Find the speed of train?
solution: Let v be the speed of train
As we know time = Distance/speed
By given condition: 360v- 360v+5=1
360v+5–360vv(v+5)=1
360(5)+ 360v -360v = v2 +5v
1800 = v2 +5v
v2 +5v- 1800=0
By Factorization method we can split 5v as -40v and 45v
v2 +5v- 18oo= v2 +45v-40v- 1800=0
v(v+45)-40(v+45)=0
(v+45)(v-40)=0
v+45=0 or v-40=0
v= -45 or v=40
speed can’t be negative
hence v= 40 km/hr.

Assignment question :
Q1) See the applications of Quadratic equation? Give two example by solving it?
Q2) Find the value of k for which the given equation has two equal roots 2x2 +kx+3=0