Quadratic Equations Master Cheat Sheet - IBPS PO (Probationary Officer) Prelims

1. IBPS PO Prelims Exam Analysis & Topic Weightage

Quadratic Equations form a vital segment of the Quantitative Aptitude section in the IBPS PO (Probationary Officer) Prelims examination. Based on historical trend analysis of NTA patterns and previous years' question papers (PYQs), candidates can expect 5 direct questions on quadratic comparisons in almost every shift.

In the IBPS PO Quantitative Aptitude section (comprising 35 questions), quadratic equations account for roughly 14% to 15% of the total weightage. Mastering rapid sign-method tricks alongside fundamental algebraic formulas detailed below allows aspirants to score a full 5/5 marks within 2 to 3 minutes.

2. Chapter 1: Standard Form & Anatomic Structure

A quadratic equation in one variable is a second-order polynomial equation expressed in its canonical representation as:

\[ax^2 + bx + c = 0\]

where \(a, b, c \in \mathbb{R}\) and \(a eq 0\).

Structural Breakdown

• Leading Coefficient (\(a\)): Must be non-zero (\(a eq 0\)). If \(a = 0\), the equation degrades into a linear equation of the form \(bx + c = 0\).
• Linear Coefficient (\(b\)): The coefficient associated with the first-degree term \(x\).
• Constant Term (\(c\)): The term independent of variable \(x\).

3. Chapter 2: The Universal Quadratic Formula

When factoring by splitting the middle term is inefficient or complex, the exact roots (solutions) of any standard quadratic equation can be universally derived using the quadratic formula:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

This algebraic formula is derived systematically by applying the method of completing the square to the standard expression \(ax^2 + bx + c = 0\).

4. Chapter 3: Discriminant (\(\Delta\)) & Nature of Roots

The expression residing under the radical sign, designated as the discriminant \(\Delta = b^2 - 4ac\), strictly governs the real and complex properties of the roots:

• Case 1: \(\Delta > 0\) — The equation possesses two distinct real roots.
• Case 2: \(\Delta = 0\) — The equation possesses two equal real roots (a repeated or coincident root given by \(x = -\frac{b}{2a}\)).
• Case 3: \(\Delta < 0\) — The equation possesses two complex conjugate roots of the form \(p \pm iq\).

5. Chapter 4: Relations Between Roots and Coefficients (Vieta's Formulas)

Let \(\alpha\) and \(\beta\) represent the two roots of the quadratic equation \(ax^2 + bx + c = 0\). According to Vieta's relations:

Sum of Roots:

\[\alpha + \beta = -\frac{b}{a}\]

Product of Roots:

\[\alpha \beta = \frac{c}{a}\]

Absolute Difference of Roots:

\[|\alpha - \beta| = \frac{\sqrt{\Delta}}{|a|}\]

6. Chapter 5: Quadratic Expressions & Parabolic Graphical Forms

A second-degree polynomial function defined as \(f(x) = ax^2 + bx + c\) traces a parabola on a two-dimensional Cartesian plane.

• Vertex Form: The polynomial can be rewritten in vertex form as \(f(x) = a(x - h)^2 + k\), where the coordinates of the vertex \((h, k)\) are defined by:

  \[(h, k) = \bigl(-\frac{b}{2a}, -\frac{\Delta}{4a}\bigr)\]

• Direction of Opening:
  • If \(a > 0\), the parabola opens vertically upward, achieving an absolute minimum value at its vertex.
  • If \(a < 0\), the parabola opens vertically downward, achieving an absolute maximum value at its vertex.

7. Chapter 6: Advanced Algebraic Manipulations & Symmetric Expressions

Forming a Quadratic Equation from Given Roots

When the roots \(\alpha\) and \(\beta\) are established, the original quadratic equation can be reconstructed as:

\[x^2 - (\text{Sum of Roots})x + (\text{Product of Roots}) = 0\]

\[x^2 - (\alpha + \beta)x + \alpha\beta = 0\]

Essential Symmetric Polynomial Identities

IBPS PO questions frequently test higher-degree symmetric manipulations involving \(\alpha\) and \(\beta\):

• \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\)
• \((\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta\)
• \(\alpha^3 + \beta^3 = (\alpha + \beta)[(\alpha + \beta)^2 - 3\alpha\beta]\)

8. People Also Ask (Frequently Asked Questions)