If the sum of the squares of the roots of the | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the roots of the equation $x^2-(a-2)x-(a+1)=0$ be $\alpha$ and $\beta$. From the relation between roots and coefficients, we have $\alpha+\bet

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(C) Let the roots of the equation $x^2-(a-2)x-(a+1)=0$ be $\alpha$ and $\beta$. From the relation between roots and coefficients, we have $\alpha+\beta = a-2$ and $\alpha\beta = -(a+1)$. The sum of the squares of the roots is given by $S = \alpha^2+\beta^2 = (\alpha+\beta)^2-2\alpha\beta$. Substituting the values, $S = (a-2)^2 - 2(-(a+1)) = (a-2)^2 + 2(a+1)$. Expanding this, $S = a^2 - 4a + 4 + 2a + 2 = a^2 - 2a + 6$. To find the minimum value, we complete the square: $S = (a^2 - 2a + 1) + 5 = (a-1)^2 + 5$. Since $(a-1)^2 \ge 0$, the minimum value of $S$ occurs when $(a-1)^2 = 0$, which implies $a = 1$.

If the sum of the squares of the roots of the equation $x^2-(a-2)x-(a+1)=0$ is least for an appropriate value of the variable parameter $a$, then the value of $a$ will be

Similar Questions

If the roots of the equation $5x^2 - 7x + k = 0$ are reciprocal to each other,then the value of $k$ is:

If the roots of the equation $12x^2 - mx + 5 = 0$ are in the ratio $2 : 3$,then $m =$

If the ratio of the roots of $x^2 + bx + c = 0$ and $x^2 + qx + r = 0$ is the same,then

If $\alpha, \beta, \gamma$ are the roots of the equation $4x^3-3x^2+2x-1=0$,then $\alpha^3+\beta^3+\gamma^3=$

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-6x^2+11x+6=0$,then $\Sigma \alpha^2 \beta+\Sigma \alpha \beta^2$ is equal to :

Vedclass Test Series

Exam Paper Generator

Online Exam Module