Solving equations is a fundamental skill for the CAT Quantitative Aptitude section. This guided path will take you from the core mechanics of linear equations to the advanced strategies for analyzing the roots of quadratic equations, building the algebraic fluency required for a top score.
Start with the fundamental skill of isolating a variable. Our interactive balancer visually demonstrates the step-by-step process of solving any linear equation.
Goal: Isolate x.
Subtract 5 from both sides to remove the constant.
Divide both sides by 3 to isolate x.
Learn the crucial skill of deconstructing a word problem and translating its language into a solvable algebraic equation.
Master the two essential algebraic methods—Substitution and Elimination—for finding a unique solution for a system of two linear equations.
Start with the core definition and use our interactive verifier to understand what the "roots" of a quadratic equation represent.
A Quadratic Equation has the form $$ ax^2 + bx + c = 0 $$
The Roots are the specific values of 'x' that make the equation true (equal to zero). Every quadratic has exactly two roots.
Learn the step-by-step process of factoring quadratic expressions to find the roots of an equation.
Goal: Rewrite $$ ax^2 + bx + c = 0 $$ into $$ (x+m)(x+n) = 0 $$.
Shortcut (when a=1): Find two numbers that multiply to 'c' and add to 'b'.
Master the universal tool for solving any quadratic equation. Our interactive calculator shows you how to apply the formula step by step.
The universal method for any quadratic:
The term inside the root, $$ b^2 - 4ac $$, is called the Discriminant.
Learn how the discriminant (b²-4ac) tells you if an equation has 2, 1, or 0 real roots without fully solving it—a critical logical shortcut.
The expression $$ b^2 - 4ac $$ is called the Discriminant. It tells us the nature of the roots without solving.
$$ a=1, b=-8, c=15 $$
$$ D = (-8)^2 - 4(1)(15) = 64 - 60 = 4 $$
Discriminant is Positive.
Result: Two distinct real roots. (Because we add/subtract a real number).
Learn the formulas for finding the sum (-b/a) and product (c/a) of the roots *without* solving the equation, a vital time-saver.
For $$ ax^2 + bx + c = 0 $$, you can find the sum and product of roots directly from coefficients.
$$ \text{Sum} = -b/a $$ | $$ \text{Product} = c/a $$
For the equation $$ 2x^2 - 10x + 12 = 0 $$
What is the SUM of the roots?
Learn the complete method for solving quadratic inequalities by finding critical points and testing regions on a number line.
Solve for x: $$ x^2 - 2x - 8 < 0 $$
Solve $$ x^2 - 2x - 8 = 0 $$
$$ (x-4)(x+2) = 0 $$
Roots: x = 4, x = -2
Test a point between -2 and 4 (e.g. 0).
$$ 0^2 - 2(0) - 8 = -8 $$
-8 < 0 is TRUE.
The parabola dips below the axis between the roots.
$$ -2 < x < 4 $$
Put your understanding of CAT Equations: A Guide to Linear, Quadratic & Higher-Order Logic to the test with our premium, adaptive practice questions. Build pacing efficiency and eliminate knowledge gaps now.
Forget formulas for a moment. Learn the 'Systematic Listing' method to build an unshakable intuition for Permutations & Combinations.
How to use 'Extreme Values' to find the range of a sum without calculating every fraction.
Stop calculating! Learn how to use the properties of symmetry to solve hard SD questions in seconds.