If a, b, c are real numbers, then both the ro | vedclass

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

Question

(C) Expanding the given equation: $(x^2 - (b+c)x + bc) + (x^2 - (a+c)x + ac) + (x^2 - (a+b)x + ab) = 0$ $3x^2 - 2(a+b+c)x + (ab+bc+ca) = 0$ The discri

Vedclass · Accepted Answer

real

Vedclass · Answer

(C) Expanding the given equation: $(x^2 - (b+c)x + bc) + (x^2 - (a+c)x + ac) + (x^2 - (a+b)x + ab) = 0$ $3x^2 - 2(a+b+c)x + (ab+bc+ca) = 0$ The discriminant $D$ of this quadratic equation is given by: $D = [-2(a+b+c)]^2 - 4(3)(ab+bc+ca)$ $D = 4(a+b+c)^2 - 12(ab+bc+ca)$ $D = 4(a^2+b^2+c^2 + 2ab+2bc+2ca - 3ab-3bc-3ca)$ $D = 4(a^2+b^2+c^2 - ab-bc-ca)$ $D = 2[(a-b)^2 + (b-c)^2 + (c-a)^2]$ Since $a, b, c$ are real, $(a-b)^2, (b-c)^2, (c-a)^2 \geq 0$. Therefore, $D \geq 0$. Since the discriminant is always non-negative, the roots are always real.

If $a, b, c$ are real numbers, then both the roots of the equation $(x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0$ are always

Similar Questions

Given $f(x) = ax^2 + bx + c$,$g(x) = a_1x^2 + b_1x + c_1$,and $p(x) = f(x) - g(x)$. If $p(x) = 0$ only for $x = -1$ and $p(-2) = 2$,what is the value of $p(2)$? Assume $a \neq a_1 \neq 0$.

What is the nature of the roots of the equation $x^2 + x = 2(x - 1)$?

If $p$ and $q$ are odd integers, then the roots of the equation $2px^{2} + (2p + q)x + q = 0$ are

If $\alpha, \beta$ are the roots of $ax^{2}+bx+c=0$ $(a \neq 0)$ and $\alpha+h, \beta+h$ are the roots of $px^{2}+qx+r=0$ $(p \neq 0),$ then the ratio of the squares of their discriminants is

The value of $k$ for which the equation $(k - 2)x^2 + 8x + k + 4 = 0$ has both roots real,distinct,and negative is

Vedclass Test Series

Exam Paper Generator

Online Exam Module