For the equation 2x^2 + 2(a + b)x + a^2 + b^2 | vedclass

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

Question

(B) Given the 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\be

Vedclass · Accepted Answer

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

Vedclass · Answer

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

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

Similar Questions

The coefficient of $x$ in the equation $x^2 + px + q = 0$ was taken as $17$ in place of $13$. Its roots were found to be $-2$ and $-15$. The roots of the original equation are:

If the harmonic mean between the roots of $(5+\sqrt{2}) x^2-b x+(8+2 \sqrt{5})=0$ is $4$,then the value of $b$ is

If one root of the equation $x^3-9x^2+26x-24=0$ is twice the other,then the sum of the cubes of those two roots is

Let $\alpha, \beta, \gamma$ be the roots of $x^3+x+10=0$ and $\alpha_1=\frac{\alpha+\beta}{\gamma^2}, \beta_1=\frac{\beta+\gamma}{\alpha^2}, \gamma_1=\frac{\gamma+\alpha}{\beta^2}$. Then,the value of $(\alpha_1^3+\beta_1^3+\gamma_1^3)-\frac{1}{10}(\alpha_1^2+\beta_1^2+\gamma_1^2)$ is

The values of $p$ for which one root of the equation $x^2 - 30x + p = 0$ is the square of the other are:

Vedclass Test Series

Exam Paper Generator

Online Exam Module