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

Exam Pattern & Weightage Analysis

In the IBPS PO (Probationary Officer) Prelims examination, the Quantitative Aptitude section consistently features 3 to 5 questions based on Quadratic Equations. Typically, candidate performance in these questions depends on speed and accuracy in comparing the roots of two separate equations (often given as variable \(x\) and variable \(y\)). Understanding the foundational algebraic principles, quick root determination methods, and the sign convention matrix drastically reduces time spent per question.

Chapter 1: Standard Form of Quadratic Equations

A quadratic equation in one variable \(x\) is a polynomial equation of the second degree. Its general standard representation is:

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

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

• \(a\): Leading coefficient (must be non-zero to maintain degree 2).
• \(b\): Linear coefficient.
• \(c\): Constant term.

Chapter 2: The Quadratic Formula & Derivation

The solutions or roots of the standard quadratic equation are determined using the fundamental quadratic formula:

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

Derivation via Completing the Square

1. Divide the entire equation by the leading coefficient \(a\): \(x^2 + \frac{b}{a}x + \frac{c}{a} = 0\)
2. Isolate the constant term to the right-hand side: \(x^2 + \frac{b}{a}x = -\frac{c}{a}\)
3. Add \(\bigl(\frac{b}{2a}\bigr)^2\) to both sides to complete the square on the left-hand side: \(\bigl(x + \frac{b}{2a}\bigr)^2 = \frac{b^2 - 4ac}{4a^2}\)
4. Take the square root of both sides and solve for \(x\) to obtain the quadratic formula.

Chapter 3: The Discriminant \(\Delta\) and Nature of Roots

The term under the square root in the quadratic formula is called the Discriminant, denoted by \(\Delta\) or \(D\):

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

The sign of \(\Delta\) dictates the real and complex properties of the roots:

• \(\Delta > 0\): Two real and distinct roots.
• \(\Delta = 0\): Two real and equal roots (a repeated single root \(x = -\frac{b}{2a}\)).
• \(\Delta < 0\): Two complex conjugate roots of the form \(p \pm iq\).

Chapter 4: Relationship Between Roots and Coefficients

Let \(\alpha\) and \(\beta\) be the roots of the quadratic equation \(ax^2 + bx + c = 0\). The relations between the roots and coefficients are given by Vieta's formulas:

• Sum of Roots: \[\alpha + \beta = -\frac{b}{a} = -\frac{\text{coefficient of } x}{\text{coefficient of } x^2}\]
• Product of Roots: \[\alpha \beta = \frac{c}{a} = \frac{\text{constant term}}{\text{coefficient of } x^2}\]
• Absolute Difference of Roots: \[|\alpha - \beta| = \frac{\sqrt{\Delta}}{|a|}\]

Chapter 5: Quadratic Function & Parabolic Graphs

Graphically, the real-valued quadratic function \(f(x) = ax^2 + bx + c\) represents a parabola in the Cartesian plane.

• If \(a > 0\), the parabola opens upward, forming a global minimum.
• If \(a < 0\), the parabola opens downward, forming a global maximum.

Vertex and Axis of Symmetry

The vertex coordinates where the maximum or minimum occurs are:

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

The vertical line representing the axis of symmetry is given by:

\[x = -\frac{b}{2a}\]

Chapter 6: Essential Algebraic Identities for Root Simplification

In competitive examinations like IBPS PO Prelims, questions frequently ask for expressions involving powers of roots \(\alpha\) and \(\beta\). Use these expanded algebraic identities:

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

Frequently Asked Questions (People Also Ask)