Question

If \[\alpha ,\,\beta \] are the roots of ${x^2} - \left( {a - 2} \right)x - \left( {a + 1} \right) = 0$ where $a$ is a variable, then the least value of ${\alpha ^2} + {\beta ^2}$ is:
(A) $2$
(B) $43$
(C) $5$
(D) $7$

Answer 1

Hint: In this question first we will find the sum of the two roots alpha and beta and then we will find the product of the two roots alpha and beta. And then with the help of this value we can easily find the value which is asked in the question by manipulation of one of the algebraic identities.

Complete step by step solution: The given equation is ${x^2} - \left( {a - 2} \right)x - \left( {a + 1} \right) = 0$ and its roots are \[\alpha \] and \[\beta \] .
We know that if \[\alpha \] and \[\beta \] are the roots of the equation $a{x^2} + bx + c = 0$ then $\alpha + \beta = - \dfrac{b}{a}$ and $\alpha \beta = \dfrac{c}{a}$ .
Now, for the equation ${x^2} - \left( {a - 2} \right)x - \left( {a + 1} \right) = 0$ we can have $b = - \left( {a - 2} \right)$ , $c = - \left( {a + 1} \right)$ and $a = 1$ .
Therefore, we can write $\alpha + \beta = - \dfrac{b}{a} = - \dfrac{{ - \left( {a - 2} \right)}}{1}$ and $\alpha \beta = \dfrac{c}{a} = \dfrac{{ - \left( {a + 1} \right)}}{1}$
We know that ${\left( {\alpha + \beta } \right)^2} = \alpha {}^2 + 2\alpha \beta + {\beta ^2}$
By manipulating the above equation. We got the equation.
$ \Rightarrow {\alpha ^2} + {\beta ^2} = {\left( {\alpha + \beta } \right)^2} - 2\alpha \beta $
Now, we can put the values $\alpha + \beta $ and $\alpha \beta $ in the above equation.
$ \Rightarrow {\alpha ^2} + {\beta ^2} = {\left( { - \left( {a - 2} \right)} \right)^2} - 2\left( { - \left( {a + 1} \right)} \right) $
$\Rightarrow {\alpha ^2} + {\beta ^2} = {a^2} - 4a + 4 + 2a + 2 $
Now, simplify the above equation
\[ \Rightarrow {\alpha ^2} + {\beta ^2} = {a^2} - 2a + 6\]
Now, the above equation can be written as
$\Rightarrow {\alpha ^2} + {\beta ^2} = {a^2} - 2a + 1 + 5$
$\Rightarrow {\alpha ^2} + {\beta ^2} = {\left( {a - 1} \right)^2} + 5$
Now, from the above equation we can say that if the value of $a$ is $1$ then we will get the minimum value of ${\alpha ^2} + {\beta ^2}$ is
$ \Rightarrow {\alpha ^2} + {\beta ^2} = {\left( {1 - 1} \right)^2} + 5 = 5$
Therefore, we got the least value ${\alpha ^2} + {\beta ^2}$ is $5$

Hence, the correct option is (C).

Note: The important thing in this question is the value of sum of the roots of the given equation and product of the roots of the equation. The other important thing in this question is the manipulation of the algebraic identity we have done in this question to find the required value. So, just be careful about these things while solving the question and also try to memorize all the algebraic identity.