If alpha and beta are the roots of the equati | vedclass

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

Question

(D) Given the quadratic equation $2x^2 - 2(m^2 + 1)x + m^4 + m^2 + 1 = 0$.Comparing this with $ax^2 + bx + c = 0$,we have $a = 2$,$b = -2(m^2 + 1)$,an

Vedclass · Accepted Answer

$m^2$

Vedclass · Answer

(D) Given the quadratic equation $2x^2 - 2(m^2 + 1)x + m^4 + m^2 + 1 = 0$.Comparing this with $ax^2 + bx + c = 0$,we have $a = 2$,$b = -2(m^2 + 1)$,and $c = m^4 + m^2 + 1$.Using the properties of roots:Sum of roots: $\alpha + \beta = -b/a = \frac{2(m^2 + 1)}{2} = m^2 + 1$ ... $(i)$Product of roots: $\alpha \beta = c/a = \frac{m^4 + m^2 + 1}{2}$ ... $(ii)$We know that $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta$.Substituting the values from $(i)$ and $(ii)$:$\alpha^2 + \beta^2 = (m^2 + 1)^2 - 2 \left( \frac{m^4 + m^2 + 1}{2} \right)$$\alpha^2 + \beta^2 = (m^4 + 2m^2 + 1) - (m^4 + m^2 + 1)$$\alpha^2 + \beta^2 = m^4 - m^4 + 2m^2 - m^2 + 1 - 1 = m^2$.

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 2(m^2 + 1)x + m^4 + m^2 + 1 = 0$,then $\alpha^2 + \beta^2$ equals:

Similar Questions

The roots of $|x - 2|^2 + |x - 2| - 6 = 0$ are

Find the number of natural number solutions for the equation $x_1 + x_2 = 100$,such that $x_1$ and $x_2$ are not multiples of $5$.

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{4} - 227 = 398$
$II.$ $y^{2} + 321 = 346$

If the sum of the roots of a quadratic equation is $1$ and the product of the roots is $-20$, find the quadratic equation.

If $x = \sqrt{7 + 4\sqrt{3}},$ then $x + \frac{1}{x} = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module