If the roots of the equations x^2 - bx + c = | vedclass

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

Question

(D) Let the roots of $x^2 - bx + c = 0$ be $\alpha$ and $\beta$, and the roots of $x^2 - cx + b = 0$ be $\alpha'$ and $\beta'$.Given that the differen

Vedclass · Accepted Answer

$-4$

Vedclass · Answer

(D) Let the roots of $x^2 - bx + c = 0$ be $\alpha$ and $\beta$, and the roots of $x^2 - cx + b = 0$ be $\alpha'$ and $\beta'$.Given that the difference between the roots is the same, we have $|\alpha - \beta| = |\alpha' - \beta'|$.For the first equation, the difference of roots is $\alpha - \beta = \sqrt{(\alpha + \beta)^2 - 4\alpha\beta} = \sqrt{b^2 - 4c}$.For the second equation, the difference of roots is $\alpha' - \beta' = \sqrt{(\alpha' + \beta')^2 - 4\alpha'\beta'} = \sqrt{c^2 - 4b}$.Equating the squares of the differences: $b^2 - 4c = c^2 - 4b$.Rearranging the terms: $b^2 - c^2 = 4c - 4b$.Factoring the left side: $(b + c)(b - c) = -4(b - c)$.Assuming $b 
eq c$ (if $b = c$, the equations are identical), we can divide by $(b - c)$ to get $b + c = -4$.

If the roots of the equations $x^2 - bx + c = 0$ and $x^2 - cx + b = 0$ differ by the same quantity, then $b + c$ is equal to

Similar Questions

Solve the given two equations and select the correct answer from the given options.
$I.$ $\quad x = \sqrt[3]{2197}$
$II.$ $\quad y^2 = 169$

If one of the roots of the equation ${x^2} + ax + 3 = 0$ is $3$ and one of the roots of the equation ${x^2} + ax + b = 0$ is three times the other root,then the value of $b$ is equal to

If the roots of $4x^2 + px + 9 = 0$ are equal,then the absolute value of $p$ is:

If $ax^2 + bx + c = 0$ and $bx^2 + cx + a = 0$ have a common root and $a, b, c$ are non-zero real numbers,then $\frac{a^3 + b^3 + c^3}{abc} = $

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

Vedclass Test Series

Exam Paper Generator

Online Exam Module