Question

If 'p' and 'q' are the roots of the equation $2a^2-4a+1=0$. Find the value of $p^3+q^3$

Answer 1

We know that if $p$ and $q$ are the roots of a quadratic equation $a{x}^2+bx+c=0$, the sum of the roots is $p+q=-\frac{b}{a}$ and the product of the roots is $pq=\frac{c}{a}$.

Here, the given quadratic equation $2a^2-4a+1=0$ is in the form $a{x}^2+bx+c=0$ where $a=2,b=-4$ and $c=1$.

The sum of the roots is:

$p+q=-\frac{b}{a}=-\frac{(-4)}{2}=2$

The product of the roots is $\frac{c}{a}$ that is:

$pq=\frac{c}{a}=\frac{1}{2}$

Now since the identity is $(a+b{)}^3=a^3+b^3+3ab(a+b)$, therefore,

$(p+q{)}^3=p^3+q^3+3pq(p+q)\Rightarrow 2^3=p^3+q^3+(3\times \frac{1}{2}\times 2)\Rightarrow 8=p^3+q^3+3\Rightarrow p^3+q^3=8-3\Rightarrow p^3+q^3=5$

Hence, the value of $p^3+q^3=5$.