Quadratic Equations for CAT: Concepts & PYQs

Quadratics and polynomial-root conditions recur and often reward structural thinking over solving roots explicitly. For CAT preparation, the useful question is not “Have I finished the chapter?” but can I recognise the version of the concept CAT hides inside a word problem and choose an efficient method under time pressure?

What to know

The core checklist for this topic is: Standard form, factorisation, discriminant, nature of roots, sum/product of roots, transformed roots; and parameter conditions, quadratic graphs. Learn these as connected ideas rather than isolated formulas. CAT often combines two of them in one question.

Recurring CAT-style forms

Expect questions built around Equal/distinct roots, integer roots, parameter range, root transformation, expression in roots, and intersection count. The wording changes, but the mathematical structure repeats. Your first 20–30 seconds should be spent identifying that structure before writing equations.

Decision rules worth remembering

• For ax²+bx+c: sum=−b/a, product=c/a, discriminant=b²−4ac
• equal roots D=0
• sign of parabola tied to leading coefficient and roots

One original example

For x²−kx+9=0 to have equal roots, discriminant must be zero: k²−36=0, so k=±6. The parameter condition is the real question; you never needed both roots.

The point of a worked example is not the answer; it is the choice of representation. After solving, ask whether a ratio table, base-100 assumption, sign chart, factorisation or diagram made the work shorter.

Common traps

• Solving roots when only sum/product needed
• missing a=0 case in parameter questions
• assuming real roots
• sign errors in Vieta

These traps are valuable because they explain why a student can “know the formula” and still lose marks. Add the exact mistake to your error log when it occurs; do not simply write “silly mistake.”

How to practise

Start with 10–15 untimed questions until the first step becomes automatic. Then use mixed timed sets where this topic is not announced in advance. Finish with sectionals and mock review. Mastery checkpoint: Can solve root-condition questions using discriminant/Vieta before reaching for quadratic formula.

Once that checkpoint is stable, move on rather than over-polishing one chapter. CAT QA is a portfolio of scoring areas; the objective is enough reliable topics to make question selection easy on exam day.