Quadratic Equations Formula Sheet and Shortcuts - IBPS PO Prelims Study Material

Welcome to this comprehensive, LibreTexts-style reference chapter on Quadratic Equations tailored specifically for the IBPS PO (Probationary Officer) Prelims examination. In the Quantitative Aptitude section of IBPS PO, Quadratic Equation comparison questions (evaluating relationships between \(x\) and \(y\)) account for 3 to 5 high-yield marks. Mastering the fundamental algebraic formulas, root-coefficient relationships, and vertex properties is essential for rapid problem-solving.

1. IBPS PO Prelims Exam Weightage & Pattern Analysis

In the IBPS PO Prelims exam, Quadratic Equations appear primarily in the Quantitative Aptitude section. Candidates are generally provided with two quadratic equations—one in terms of variable \(x\) and another in terms of variable \(y\)—and are required to establish the mathematical relationship between the two variables (e.g., \(x > y\), \(x \ge y\), \(x < y\), \(x \le y\), or \(x = y\) / Relationship Cannot Be Established).

• Expected Questions: 3–5 Questions
• Target Time per Question: 30–45 seconds
• Core Strategy: Utilize sum/product of roots and sign shortcuts to determine root values without executing full factorization.

2. Chapter 1: Standard Form & Fundamentals

A quadratic equation in one variable is a polynomial equation of degree 2. Its standard form is represented as:

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

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

Key Components of the Standard Equation:

• Leading Coefficient (\(a\)): Determines the width and direction of the parabolic curve (opens upwards if \(a > 0\), downwards if \(a < 0\)).
• Linear Coefficient (\(b\)): Influences the position of the axis of symmetry and the slope of the curve.
• Constant Term (\(c\)): Represents the y-intercept of the function where \(x = 0\).

3. Chapter 2: The Universal Quadratic Formula

When factorizing a quadratic equation by splitting the middle term becomes time-consuming or difficult due to large coefficients, the universal quadratic formula is used to extract the exact roots \(x_1\) and \(x_2\):

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

Derived via the technique of completing the square, this equation systematically yields real or complex solution sets for any quadratic polynomial.

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

The expression residing under the radical sign in the quadratic formula is termed the discriminant, denoted by \(\Delta\) (or \(D\)):

\[\Delta = b^2 - 4ac\]

The value of the discriminant determines the nature and existence of the roots:

• When \(\Delta > 0\): The equation possesses two distinct real roots.
• When \(\Delta = 0\): The equation possesses exactly one real repeated root (double root), where \(x_1 = x_2 = -b / (2a)\).
• When \(\Delta < 0\): The equation has no real roots; instead, it yields two complex conjugate roots.

5. Chapter 4: Relations Between Roots and Coefficients

Understanding Vieta's formulas relating the roots (\(x_1\) and \(x_2\)) to the original coefficients (\(a, b, c\)) allows candidates to quickly check conditions in IBPS PO comparison questions without finding exact roots.

• Sum of Roots: \[x_1 + x_2 = -\frac{b}{a}\]
• Product of Roots: \[x_1 x_2 = \frac{c}{a}\]
• Absolute Difference of Roots: \[|x_1 - x_2| = \frac{\sqrt{\Delta}}{|a|}\]

6. Chapter 5: Vertex Form & Parabolic Symmetry

Any quadratic function \(f(x) = ax^2 + bx + c\) can be transformed into its vertex form through algebraic completion of squares:

\[f(x) = a(x - h)^2 + k\]

Geometrical Parameters:

• Vertex Coordinates \((h, k)\): \(h = -\frac{b}{2a}\) and \(k = f\bigl(-\frac{b}{2a}\bigr)\).
• Axis of Symmetry: The vertical line \(x = h = -\frac{b}{2a}\).

7. Chapter 6: Key Algebraic Identities for Speed Math

To compute combinations of roots rapidly during exam pressure, memorize these derived identities involving sum (\(x_1 + x_2\)) and product (\(x_1 x_2\)):

• Sum of Squares of Roots: \[x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1x_2\]
• Square of Root Difference: \[(x_1 - x_2)^2 = (x_1 + x_2)^2 - 4x_1x_2\]
• Sum of Cubes of Roots: \[x_1^3 + x_2^3 = (x_1 + x_2)(x_1^2 - x_1x_2 + x_2^2)\]

8. People Also Ask (Frequently Asked Questions)