Question

If (2x+1) and (x-3)are the factors of ax2 - 5x +c then the value of A and C are respectively A) 2,3 B) - 2,3 C) 2,-3 D) 1, - 3

Answer 1

-2,3
b)-2,3
c)2-3
d)1,-3

Answer 2

let f(x)=ax2-5x+c
given the factors of f(x) as 2x+1 and x-3
we know that zeroes of the polynomial as 2x+1=0,i.e;x=-1/2 and x-3=0,i.e;x=3
also if we substitute the zeroes of f(x) in f(x) the result will be zero
therefore f(-1/2)=ax2-5x+c=0
f(-1/2)=a(-1/2)2-5(-1/2)+c=0
f(-1/2)=a(1/4)+5/2+c=0
f(-1/2)=a/4+5/2+c=0
f(-1/2)=a+10+4c/4=0
f(-1/2)=a+10+4c=0
a+4c=-10_______(1)
f(3)=ax2-5x+c=0
f(3)=a(3)2-5(-3)+c=0
f(3)=a(9)+15+c=0
f(3)=9a+15+c=0
9a+c=-15____________(2)
multiply equation (2) with 4 inorder to cancel c from the equation
after multiplying with 4 in the second equation
36a+4c=-60__________(2)
from (1)&(2)
a+4c=-10
36a+4c=-60
_-___-___+___
-35a=50
a=-50/35
therefore a=-10/7
substitute a value in (1)
a+4c=-10
-10/7+4c=-10
-10/7+28c/7=-10
-10+28c=-70
now take 2 as common
then -2(5-14c)=-2(35)
now cancel both -2s from lhs as well as rhs
then 5-14c=35
-14c=35-5
-14c=30
c=-30/14
c=-15/7
therefore a=-10/7,c=15/7