Determine K and solve the equation x^3-kx^2+3x+54=0, if one of its zeros is triple of another.
let the roots be a, 3a and b
then
(x-a)(x-3a)(x-b) = x^3 - kx^2 + 3x + 54
looking at the last term -3a^2b = 54
a^2b = -18 , the only square factors of -18 are 1 and 9
case 1: a=1, b = -18
then
(x-1)(x-3)(x+18) would be the expression
this give us x^3 + 14x^2 - 69x + 54
which would not match up the x terms
case 2: a=3, b=-2
then
(x-3)(x-9)(x+2) would be the expression for
x^3 - 10x^2 + 3x + 54
this matches if k = 10
and the roots are 3,9 and -2
(there should be an easier way to do this)
To determine the value of K and solve the equation x^3 - kx^2 + 3x + 54 = 0, when one of its zeros is triple the value of another, we can use Vieta's formulas.
Let's assume the zeros of the equation are a, b, and 3b (where b is non-zero).
Vieta's formulas state that for a cubic equation of the form ax^3 + bx^2 + cx + d = 0, the sum of the roots is given by:
Sum of roots (a + b + 3b) = -b/a
And the product of the roots is given by:
Product of roots (a * b * 3b) = -d/a
In our equation, the sum of the roots is a + b + 3b = -(-k)/1 = k
And the product of the roots is a * b * 3b = 54/1 = 54
We have two equations:
1) k = a + b + 3b
2) 54 = a * b * 3b
Now we can solve these equations to find the value of K and the corresponding roots.
Substituting the value of k from equation 1 into equation 2, we get:
54 = (k - b - 3b) * b * 3b
54 = (k - 4b) * b^2 * 3
Simplifying, we have:
18 = (k - 4b) * b^2
Dividing both sides by b^2:
18/b^2 = k - 4b
Rearranging the equation:
k = 18/b^2 + 4b
Now we have the equation for K in terms of b. We can substitute this value of K back into equation 1 to find the value of b:
k = a + b + 3b
Substituting k = 18/b^2 + 4b:
18/b^2 + 4b = a + b + 3b
Rearranging the equation and simplifying:
18/b^2 = a + 2b
Now, we can choose a value for b and find the corresponding values of a and K, and then solve for the roots of the equation.
For example, let's say b = 2:
k = 18/2^2 + 4*2 = 18/4 + 8 = 4.5 + 8 = 12.5
a = 18/2^2 - 2*2 = 18/4 - 4 = 4.5 - 4 = 0.5
Hence, when b = 2, we have a = 0.5 and k = 12.5.
To find the roots of the equation, we can substitute these values of a and k back into the original equation:
x^3 - kx^2 + 3x + 54 = 0
x^3 - 12.5x^2 + 3x + 54 = 0
You can use numerical methods or the Rational Root Theorem to find the roots of this equation.