TS EAMCET · Maths · Quadratic Equation

If the quadratic equations $x^{2} - 7 x + 3 c = 0$ and $x^{2} + x - 5 c = 0$ have a common root, then for non-zero real value of $c$ the sign of the expression $x^{2} - 3 x + c$ is

A negative for all $x \in ℝ$
B positive for all $x \in (1, 3)$
C negative for all $x \in (1, 3)$
D positive for all $x \in ℝ$

Verified Solution

Answer & Solution

Correct Answer

(D) positive for all $x \in ℝ$

Step-by-step Solution

Detailed explanation

Given, x2-7x+3c=0 and x2+x-5c=0 have a common root, So assuming α to be common root then, α2-7α+3c=0 ...1 α2+α-5c=0 ......2 Solving equation 1 & 2 we get, α235c-3c=-α-5c-3c=11+7 ⇒α=c & α2=4c…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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If the quadratic equations x2-7x+3c=0 and x2+x-5c=0 have a common root, then for non-zero real value of c the sign of the expression x2-3x+c is

Step-by-step Solution

If the quadratic equations $x^{2} - 7 x + 3 c = 0$ and $x^{2} + x - 5 c = 0$ have a common root, then for non-zero real value of $c$ the sign of the expression $x^{2} - 3 x + c$ is