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 can use the sum and product of the roots of a quadratic equation.
Let's first write the given cubic equation in standard form:
ax^3 - 6x^2 + 2ax - 3b = 0
Now, let's find the sum of the roots of this equation. According to Vieta's formulas, for a cubic equation in the form ax^3 + bx^2 + cx + d = 0, the sum of the roots (denoted as S) is given by:
S = -b / a
In our case, the sum of the roots is 3. So, we have:
S = -(-6) / a = 3
Simplifying this equation:
6 / a = 3
Multiplying both sides by a:
6 = 3a
Dividing by 3:
a = 2
Next, let's find the product of the roots. According to Vieta's formulas, the product of the roots (denoted as P) is given by:
P = (-1)^n * d / a
where n is the degree of the equation. In our case, n = 3. So, we have:
P = (-1)^3 * (-3b) / a = 3
Simplifying this equation:
3b / 2 = 3
Multiplying both sides by 2:
3b = 6
Dividing by 3:
b = 2
Therefore, the values of a and b are 2 and 2, respectively.