Quant
Time, Speed & Distance
Every CAT Time, Speed & Distance formula on one page — relative speed, trains crossing poles and platforms, boats and streams, and delay problems — each with a worked example.
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Linear & Quadratic Equations
CAT'26 QUANT CHEATSHEET
Every linear and quadratic equation formula you need for CAT 2026 — on one page.
Linear and quadratic equations form the algebraic backbone of CAT quant, and the marks here come from speed, not difficulty. A system of two linear equations, a discriminant check, or a quadratic inequality should all take under a minute once the pattern is recognised. This sheet lays out every formula you need for the CAT quant section: solving linear systems by ratio comparison, fractional equations, parameter questions, the discriminant and quadratic formula, sum and product of roots, nature-of-roots logic, and the maxima-minima and graph shortcuts that avoid completing the square by hand, each with a worked example in real numbers. Keep it open while you practise, and after a mock check where you stand on the CAT score predictor to see which idea is costing you marks.
Dev note: the canonical route
/cheatsheets/optima-learn-linear-quadratic-equations-cheatsheet is not live yet
(no /cheatsheets hub in the current sitemap). Built production-ready as a drop-in once it ships.Linear & Quadratic Equations: every formula you need
1Linear Equation Basics
One equation, one unknown, one solution.
ax + b = 0 → x = −b/a (a ≠ 0)
Example: 5x − 15 = 0 → x = 3.
CAT Hack: You need n independent equations to uniquely solve n unknowns.
2System of Linear Equations
The ratio of coefficients tells you the solution type.
Unique: a1/a2 ≠ b1/b2 | Infinite / No solution: equal or clash with c1/c2
Example: 3x+2y=12, 5x−2y=20 → x=4, y=0.
CAT Hack: Need x + y? Don't solve individually — add or subtract the equations directly.
3Fractional Linear Equations
Clear denominators first, then solve like normal.
Find LCM ↓ multiply every term ↓ solve ↓ check denominator ≠ 0
Example: x/3 + x/4 = 14, LCM = 12 → 4x+3x=168 → x = 24.
4Parameter Questions
Find k so the system has a specific solution type.
Unique: a1/a2 ≠ b1/b2 | Infinite: a1/a2 = b1/b2 = c1/c2
Example: kx+3y=7, 4x+6y=14, infinite solutions → k = 2.
5Integer Solutions
Count non-negative or positive integer solutions to a linear equation.
non-negative solutions of x1+…+xr=n: n+r−1Cr−1
Example: x+y+z=10, non-negative → 12C2 = 66.
CAT Insight: For strictly positive integers, use (n−1)C(r−1) instead of the non-negative form.
6Linear Inequalities
Multiplying or dividing by a negative flips the sign.
+c or −c: sign unchanged | ×(−ve): flip the sign
Example: −2x+5 ≤ 11 → −2x ≤ 6 → x ≥ −3.
Common Mistake: Always flip the inequality sign after multiplying or dividing by a negative number.
7Quadratic Basics
Three equivalent ways to write a quadratic.
Standard: ax2+bx+c=0 | Factored: a(x−r1)(x−r2)=0
Example: x2−6x+8=0 = (x−2)(x−4)=0 → x = 2, 4.
8Discriminant
One number tells you how many real roots exist.
D = b2−4ac | D>0: 2 real roots · D=0: equal · D<0: none real
Example: x2+4x+4: D = 16−16 = 0 → repeated root.
9Sum & Product of Roots
Read off both without solving for the roots.
α+β = −b/a | αβ = c/a
Example: x2−5x+3=0 → α+β=5, αβ=3.
10Factorisation
Split the middle term using product and sum of roots.
find two numbers: product = ac, sum = b → split → group → factor
Example: 6x2+x−2 = (2x−1)(3x+2) → x = ½, −⅔.
11Quadratic Formula
The direct route to both roots.
x = (−b ± √D) / 2a (real roots need D ≥ 0)
Example: 2x2−5x+1=0 → x = (5±√17)/4.
CAT Hack: Compute D first — it tells you whether real roots even exist before you commit to the formula.
12Equation from Roots
Build the quadratic back up from its roots.
x2−(α+β)x+αβ=0 | reciprocal roots: swap a and c
Example: roots 3 and −5 → x2+2x−15=0.
13Nature & Sign of Roots
The signs of the sum and product reveal the root signs.
Both +ve: −b/a<0, c/a>0 | Both −ve: −b/a>0, c/a>0 | Opposite: c/a<0
Example: x2−7x+12: α+β=7, αβ=12 → roots 3, 4 (both positive).
14Quadratic Inequalities
For a > 0, the sign flips between the roots.
a>0: f(x)>0 outside roots, f(x)<0 between | a<0: signs reverse
Example: x2−5x+6>0, roots 2 and 3 → x<2 or x>3.
15Maxima & Minima
The vertex gives the turning point value directly.
vertex at x = −b/2a | max/min value = (4ac−b2)/4a
Example: x2−6x+14 → minimum = 5, at x = 3.
CAT Favourite: Vertex-form questions are a CAT favourite — the (4ac−b²)/4a shortcut skips completing the square.
16Parabola & Graph
The sign of a decides which way the parabola opens.
a>0: opens up | a<0: opens down | vertex at x=−b/2a
Example: x2−6x+14 has a=1>0 → opens upward, vertex at x=3.
CAT exam shortcuts & traps
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CAT Exam Shortcuts
- Add or subtract equations directly instead of solving for each variable
- System type: compare a1/a2, b1/b2, c1/c2
- α+β = −b/a; αβ = c/a — no need to solve for the roots
- Compute the discriminant D first before applying the quadratic formula
- Vertex: x = −b/2a; turning-point value = (4ac−b2)/4a
- Quadratic inequality (a>0): outside the roots is positive, between is negative
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Most Common CAT Traps
- Forgetting to flip the inequality sign when multiplying or dividing by a negative.
- Solving a system variable-by-variable instead of adding/subtracting the equations.
- Applying the quadratic formula without checking the discriminant first.
- Confusing sum of coefficients or root signs: c/a<0 means the roots have opposite signs, not that both are negative.
- Mixing up greatest coefficient logic with quadratic inequality sign rules.
This topic rewards recognising the setup over grinding through algebra — comparing coefficient ratios or reading off the discriminant is almost always faster than solving from scratch. Drill this sheet until the sum-product shortcuts and the discriminant check are reflex, then test them on full sets and track progress with the CAT score predictor. For more topic guides, browse the Optima Learn blog or explore every study guide, and work through the full CAT exam hub for section-wise strategy. When you want structured, mentor-led prep, the team at Optima Learn can map out your plan — book a free CAT 2026 call and line up your next eight weeks.
