if the sum of the roots and product of the roots of ax^3-6x2+2ax-3b=0 is 3&6 respectively.Then find a and b.
To find the values of a and b, we need to use Vieta's formulas, which relate the coefficients of a polynomial to its roots.
For a cubic equation in the form of ax^3 + bx^2 + cx + d = 0, Vieta's formulas state that:
1. The sum of the roots is equal to -b/a.
2. The product of the roots is equal to -d/a.
In the given equation: ax^3 - 6x^2 + 2ax - 3b = 0, the sum of the roots is given as 3, and the product of the roots is given as 6.
So, we have the following equations:
1. -6/a = 3 [Sum of roots]
2. -3b/a = 6 [Product of roots]
Let's solve these equations step by step to find the values of a and b.
Equation 1: -6/a = 3
To find 'a', we can cross-multiply and solve for it:
-6 = 3a
Divide both sides by 3:
-2 = a
So, we have found the value of 'a' to be -2.
Equation 2: -3b/a = 6
Now, substituting the value of 'a' as -2, we have:
-3b/(-2) = 6
To find 'b', we can cross-multiply and solve for it:
-3b = 6 * -2
-3b = -12
Divide both sides by -3:
b = 4
Therefore, the values of 'a' and 'b' are -2 and 4, respectively.
Hence, a = -2, and b = 4.