Find the values of h and k for which x=-2 and x=3/4 are solution of the equation hx²-kx-6
Answers
Step-by-step explanation:
here the solutions of the equation are x = -2 , x = 3/4
let the solutions are α and β ,
α = -2
β = 3/4
given equation is hx² - kx - 6 , compare it with standard form of quadratic equation ax² + bx + c
therefor, a = h b = -k c = -6
we know the relation between coefficients and roots of quadratic equation
α + β = -b / a and α.β = c/a
α.β = (-6)/h
(-2)(3/4) = (-6) / h
-(3/2) = (-6)/h
(-3/2)(1/-6)= 1/h
(-3 / (-12)) =1/ h
1/4 = 1/h
h = 4
Now,
α + β = -b / a
(-2) + (3/4) = - (-k) / (h)
put the value of h
(-2) + (3/4) = k / 4
( ((-8) + 3) / 4) = k/4
4 ( ((-8) + 3) / 4) = k
(-8) + 3 = k
k = - 5
values of h and k are 4 and -5
verification :
put the values of h, k and the values of x in a given equation and equate it with zero
hx²-kx-6
(4)(-2)² - (-5)(-2) - 6 = 0
(4)(4) - 10 - 6 = 0
16 - 16 = 0
0 = 0
Now, put x = 3/4
(4)(3/4)² -(-5)(3/4) - 6 = 0
(4)(9/16)+ (15 / 4) - 6 = 0
(9 / 4) + (15 /4) - 6 =0
(24 / 4) - 6 =0
6 - 6 = 0
hence verified .