The value of p for which the sum of the squar | vedclass

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

Question

(B) Let the roots of the quadratic equation be $\alpha$ and $\beta$. According to the properties of roots,$\alpha + \beta = p + 3$ and $\alpha \beta =

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(B) Let the roots of the quadratic equation be $\alpha$ and $\beta$. According to the properties of roots,$\alpha + \beta = p + 3$ and $\alpha \beta = 5p - 2$. The sum of the squares of the roots is given by $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta$. Substituting the values,we get $\alpha^2 + \beta^2 = (p + 3)^2 - 2(5p - 2)$. Expanding this,$\alpha^2 + \beta^2 = p^2 + 6p + 9 - 10p + 4 = p^2 - 4p + 13$. To find the minimum value,we complete the square: $p^2 - 4p + 13 = (p^2 - 4p + 4) + 9 = (p - 2)^2 + 9$. The expression $(p - 2)^2 + 9$ attains its minimum value when $(p - 2)^2 = 0$,which implies $p = 2$.

The value of $p$ for which the sum of the squares of the roots of the equation $x^2 - (p + 3)x + (5p - 2) = 0$ assumes its least value is:

Similar Questions

If the sum of the roots of the quadratic equation $ax^2 + bx + c = 0$ is equal to the sum of the squares of their reciprocals,then $\frac{b^2}{ac} + \frac{bc}{a^2} = $

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

If $\alpha$ and $\beta$ are the roots of the equation $x^2 + 6x + \lambda = 0$ and $3\alpha + 2\beta = -20$,then $\lambda = $

If the coefficients of the equation whose roots are $k$ times the roots of the equation $x^3+\frac{1}{4} x^2-\frac{1}{16} x+\frac{1}{144}=0$ are integers,then a possible value of $k$ is

If $\alpha, \beta$ are the roots of the equation $Ax^2 + Bx + C = 0$ and $\alpha^2, \beta^2$ are the roots of the equation $x^2 + px + q = 0$,then $p = \dots$

Vedclass Test Series

Exam Paper Generator

Online Exam Module