Question

If $\alpha $, $\beta $ are the roots of the equation ${{x}^{2}}+ax+b=0$ then the value of ${{\alpha }^{3}}+{{\beta }^{3}}$ is equal to
A. $-({{a}^{3}}+3ab)$
B. ${{a}^{3}}+3ab$
C. $-{{a}^{3}}+3ab$
D. ${{a}^{3}}-3ab$

Answer 1

Hint: In this question, we are to find the sum of the cubes of the roots of the given quadratic equation. For this, we use the sum and product of the roots of a quadratic equation formula. By simply applying the coefficients of the given equation into the formulae, we get the value of the required expression.

Formula UsedConsider a quadratic equation $a{{x}^{2}}+bx+c=0$.
Then, its roots are $\alpha $ and $\beta $. So, the sum of the roots and the product of the roots is formulated as:
$\alpha +\beta =\dfrac{-b}{a};\alpha \beta =\dfrac{c}{a}$
General cubes formula we have
${{x}^{3}}+{{y}^{3}}=(x+y)({{x}^{2}}-xy+{{y}^{2}})$



Complete step by step solution:The given quadratic equation is ${{x}^{2}}+ax+b=0$ and its roots are $\alpha $ and $\beta $.
So, comparing the given equation with the standard equation, we get
$a=1;b=a;c=b$
Then, the sum of the roots of the given equation is obtained as
$\alpha +\beta =\dfrac{-a}{1}=-a\text{ }...(1)$
And the product of the roots of the given equation is
$\alpha \beta =\dfrac{b}{1}=b\text{ }...(2)$
The required expression with the roots of the given equation is ${{\alpha }^{3}}+{{\beta }^{3}}$
On simplifying, we get
$\begin{align}
  & {{\alpha }^{3}}+{{\beta }^{3}}=(\alpha +\beta )({{\alpha }^{2}}-\alpha \beta +{{\beta }^{2}}) \\
 & \text{ }=(\alpha +\beta )({{\alpha }^{2}}-\alpha \beta +{{\beta }^{2}}+2\alpha \beta -2\alpha \beta ) \\
 & \text{ }=(\alpha +\beta )({{\alpha }^{2}}+2\alpha \beta +{{\beta }^{2}}-3\alpha \beta ) \\
 & \text{ }=(\alpha +\beta )\left[ {{(\alpha +\beta )}^{2}}-3\alpha \beta \right] \\
\end{align}$
Substituting (1) and (2) values, in the above expression, we get
$\begin{align}
  & {{\alpha }^{3}}+{{\beta }^{3}}=(\alpha +\beta )\left[ {{(\alpha +\beta )}^{2}}-3\alpha \beta \right] \\
 & \text{ }=(-a)\left[ {{(-a)}^{2}}-3b \right] \\
 & \text{ }=-{{a}^{3}}+3ab \\
\end{align}$
Thus, the required expression’s value is $-{{a}^{3}}+3ab$.


option C is correct

Note: Here, we need to remember the sum and product formulae. So, we can substitute these values in the cube formula to find the given expression. And be sure with the signs in the formulae. Otherwise, the entire calculation may go in the wrong way.