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

(B) Given the quadratic equation $2x^2 + 2(a + b)x + a^2 + b^2 = 0$.Sum of roots $\alpha + \beta = -\frac{2(a + b)}{2} = -(a + b)$.Product of roots $\

Vedclass · Accepted Answer

$x^2 - 4abx - (a^2 - b^2)^2 = 0$

Vedclass · Answer

(B) Given the quadratic equation $2x^2 + 2(a + b)x + a^2 + b^2 = 0$.Sum of roots $\alpha + \beta = -\frac{2(a + b)}{2} = -(a + b)$.Product of roots $\alpha\beta = \frac{a^2 + b^2}{2}$.Now,$(\alpha + \beta)^2 = (-(a + b))^2 = (a + b)^2 = a^2 + b^2 + 2ab$.Also,$(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta = (a + b)^2 - 4\left(\frac{a^2 + b^2}{2}\right) = a^2 + b^2 + 2ab - 2a^2 - 2b^2 = 2ab - a^2 - b^2 = -(a - b)^2$.The required equation is $x^2 - [(\alpha + \beta)^2 + (\alpha - \beta)^2]x + [(\alpha + \beta)^2 \cdot (\alpha - \beta)^2] = 0$.Sum of new roots: $(a + b)^2 - (a - b)^2 = (a^2 + b^2 + 2ab) - (a^2 + b^2 - 2ab) = 4ab$.Product of new roots: $(a + b)^2 \cdot (-(a - b)^2) = -((a + b)(a - b))^2 = -(a^2 - b^2)^2$.Thus,the equation is $x^2 - 4abx - (a^2 - b^2)^2 = 0$.

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 + 2(a + b)x + a^2 + b^2 = 0$,then the equation whose roots are $(\alpha + \beta)^2$ and $(\alpha - \beta)^2$ is

Similar Questions

If one root of $x^3-7x^2+36=0$ is twice the other,then the sum of those two roots is

If the roots of $x^2 - 7x + 6 = 0$ are $\alpha$ and $\beta$,then find the value of $\frac{1}{\alpha} + \frac{1}{\beta}$.

In a triangle $PQR$,$\angle R = \pi / 2$. If $\tan(P/2)$ and $\tan(Q/2)$ are the roots of the quadratic equation $ax^2 + bx + c = 0$,where $a \neq 0$,then which of the following is true?

Let $p$ and $q$ be two positive numbers such that $p + q = 2$ and $p^{4} + q^{4} = 272$. Then $p$ and $q$ are roots of the equation:

If $\alpha^2 = 5\alpha - 3$ and $\beta^2 = 5\beta - 3$ where $\alpha \neq \beta$,what is the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module