Given that the quadratic equation hx^2 - 4kx + 9h =0 has two equal roots, find the ratio h:k
The answer given is 4:6. I need to know the method pls... Thx in advance
using the properties dealing with the nature of the roots,
if there are two equal roots, then b^2 - 4ac = 0
so in hx2 _ 4kx + 9h = 0
16k^2 -4(h)(9h) = 0
16k^2 = 36h^2
4k = 6h
h/k = 4/6
So h : k = 4 : 6 = 2 : 3
They should have reduced their answer to 2:3
To find the ratio between h and k, we can use the fact that the quadratic equation has two equal roots.
First, let's simplify the quadratic equation hx^2 - 4kx + 9h = 0. We can divide the equation by h to make the leading coefficient 1:
x^2 - (4k/h)x + 9 = 0
Since the quadratic equation has two equal roots, we know that the discriminant (b^2 - 4ac) must be equal to zero.
In this case, a = 1, b = -4k/h, and c = 9. The discriminant is given by:
D = (b^2 - 4ac)
Substituting the values:
0 = (-4k/h)^2 - 4(1)(9)
0 = 16(k^2/h^2) - 36
Now, let's solve this equation for the ratio h:k:
16(k^2/h^2) = 36
Multiplying both sides by h^2:
16k^2 = 36h^2
Dividing both sides by 4:
4k^2 = 9h^2
Taking the square root of both sides:
2k = 3h
Rearranging the equation:
h/k = 2/3
To find the ratio h:k, we multiply both sides by 3:
3(h/k) = 2
Simplifying further:
h/k = 2/3
So, the ratio h:k is 2:3.
However, in the answer given (4:6), all the terms are multiplied by a common factor of 2. Thus, the ratio can be simplified further:
h:k = 2(2):2(3)
h:k = 4:6
Therefore, the ratio h:k is indeed 4:6.