Algebra (Linear/Quadratic Equations, Progressions)

Syllabus Unit: Quantitative Ability (QA)

Why it matters

Chapter overview

CAT algebra tests equation-solving speed, sign discipline with inequalities and modulus, fluency with AP/GP/HP and the AM-GM-HM inequality chain, and reading functions and logarithms as tools rather than abstract objects — together these form the backbone of harder QA sets and DILR-adjacent quant reasoning.

Management Aptitude Syllabus Quantitative Ability (QA) 34% weightage

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1Identify a, b, cFrom ax²+bx+c=0

2Compute DiscriminantD = b²−4ac

3Apply Vieta'sSum=−b/a, Product=c/a

4Interpret RootsReal/equal/complex, signs

1Linear & Quadratic Equations

Vieta's formulas let you find sum and product of roots without ever solving for the roots individually.

Learning Objectives

Solve simultaneous linear equations using substitution/elimination and recognise when a system has no or infinite solutions.
Apply Vieta's formulas: for ax²+bx+c=0, sum of roots = −b/a, product of roots = c/a.
Use the discriminant b²−4ac to determine whether roots are real-distinct, real-equal, or complex.
Form a quadratic equation given conditions on its roots (sum, product, or a stated ratio).

Key Terms

DiscriminantD = b²−4ac for ax²+bx+c=0; D>0 gives two real distinct roots, D=0 gives equal roots, D<0 gives complex roots.

Vieta's FormulasFor a quadratic ax²+bx+c=0 with roots α,β: α+β = −b/a and αβ = c/a.

Consistent SystemA system of linear equations that has at least one solution; inconsistent systems (parallel lines) have none.

Quadratic Formulax = (−b ± √(b²−4ac)) / 2a, giving both roots of ax²+bx+c=0 directly.

CAT rarely asks you to just 'solve the quadratic' — it asks about properties of the roots (sum, product, sign, nature), which **Vieta's formulas** let you extract without ever finding the actual roots.

For ax² + bx + c = 0: Sum of roots = −b/a, Product of roots = c/a, Discriminant D = b²−4ac

Worked example: roots in a given ratio

For x²−7x+k=0 with roots in ratio 2:5, set roots as 2m and 5m. Sum = 7m = 7 (from −b/a = 7) so m=1; product = 10m² = 10 = k.

Key PointIf D<0, the quadratic has no real roots — the graph never touches the x-axis. CAT sometimes asks 'for what value of k does the equation have real roots', which is a discriminant ≥ 0 inequality.

Simultaneous linear equations

Unique solution: lines intersect at one point (different slopes).
No solution: lines are parallel (same slope, different intercept).
Infinite solutions: lines are identical (same slope and intercept).

Exam TipA quadratic with rational coefficients and one irrational root (e.g. 2+√3) must have its conjugate (2−√3) as the other root — CAT tests this to construct equations from a single given root.