Question:
The value of $a$ for which the sum of the squares of the roots of the equation $x^2 - (a - 2) x - a - 1 = 0$ assume the least value is
The value of $a$ for which the sum of the squares of the roots of the equation $x^2 - (a - 2) x - a - 1 = 0$ assume the least value is
Updated On: Jul 5, 2022
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The Correct Option is A
Solution and Explanation
Let $\alpha, \beta$ be the roots of the equation
$\therefore\, \alpha+\beta=a-2$ and $\alpha\beta=-\left(a+1\right)$
Now $\alpha^{2}+\beta^{2}=\left(\alpha+\beta\right)^{2}-2\alpha\beta$
$=\left(a-2\right)^{2}+2\left(a+1\right)$
$=\left(a-1\right)^{2}+5$
$\therefore\, \alpha^{2}+\beta^{2}$ will be minimum if $\left(a-1\right)^{2}=0$ ,
i.e., $a=1$ .
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
