Introduction to Algebra | Quantitative for GMAT PDF Download

GMAT algebra questions cover almost 16.3% of thequantitative reasoning section.Approximately, 15 questions appear from algebra, so you need to prepare for this section with regular practice sessions. 
Basic GMAT algebra concepts include Inequalities, Functions, Quadratic, and Linear Equations. Remember that you are not allowed to use any form of calculator to answer your questions.

Now, we will first study Algebraic terms, Algebraic expressions, and Algebraic concepts respectively to build a good conceptual knowledge.

Basics of Algebra

• Variables 
  They are the symbols that represent the variable integers in GMAT algebra. 
  x and y are variables in the equation 9x + 4y + 2 as their value is not defined.
• Constants
  Constants are the values that do not change in algebraic problems. 
  2 is a constant term in equation 9x + 4y + 2 as it has one unique and defined value.
• Terms
  These are formed when both the variables and constants are combined in the algebraic concepts. 
  For example, in the expression 9x + 4y + 2: 9x, 4y, and 2 are three terms.
• Algebraic Expression
  GMAT algebra formulas like (x + 2), (x – 3c), and (2x – 3y) are all examples of algebraic expressions, which have terms jumbled together by addition or deduction.
• Degree
  The power of the variables is measured by its degrees. 
  The highest power of the variable is called a degree.
• Coefficient 
  It refers to a number or a symbol that is used to indicate a property or a set of properties. 
  In the expression 9x + 4y + 2, Coefficient of x = 9 and Coefficient of y = 4

Key Difference at a Glance

• GMAT algebra concepts include inequalities that work with non-equal comparisons between 2 numbers or other mathematical concepts. 
• The fundamental concepts are - Mathematical operations with inequalities, Functioning with ranges of numbers, Transitive property, and Addition of like Qualities. 
• The four signs of GMAT inequality are -

  Inequality sign	Variables x and y	interpretations
  >	x > y	x is greater than y
  <	X < y	x is lesser than y
  ≥	x ≥ y	x is greater than or equal to y
  ≤	x ≤ y	x is lesser than or equal to y

• One of the sample questions from inequalities for your understanding is given below –
• Example: 

• A function (f(x)  is defined by an equation or rule that takes a value (often denoted as ( x )) as input and produces a unique result, or output, based on that input. The notation ( f(x)  is read as "f of x" and represents the output value when the input is ( x ).
  

• Types of functions that are commonly encountered in GMAT are written below

• 1. Linear Functions
  Represented by  f(x) = mx + b , where ( m ) is the slope and ( b ) is the y-intercept. 
  These functions graph as straight lines and describe relationships with a constant rate of change.
  Example:  f(x) = 3x + 5 

• 2. Quadratic Functions
  Represented by ( f(x) = ax² + bx + c ), where ( a ), ( b ), and ( c ) are constants. 
  Quadratic functions graph as parabolas and are common in GMAT questions involving areas, velocities, or maximizing/minimizing values.
  Example:  f(x) = x² - 4x + 3 

• 3. Exponential Functions

  Represented by  f(x) = a b^x, where ( a ) and ( b ) are constants. 
  Exponential functions show rapid increases or decreases and often appear in growth or decay problems.
  Example:  f(x) = 2 .3^x 

• 4. Absolute Value Functions
  Represented by  f(x) = |x| 
  Absolute value functions measure distance from zero and are always non-negative.
  Example: f(x) = |x - 5| 

• Quadratic equations are another concept in the GMAT algebra syllabus and are considered one of the complex parts of algebraic expressions. 

• It deals with the factoring method of quadratic equations, determining solutions for the Difference of the Perfect squares, Root, and quadratic formula. 

• Roots of a quadratic equation

  The roots of the quadratic equation are nothing but the solutions of the equations. A quadratic equation can have a maximum of two roots - it can have 0 to 1 or 2 roots.The roots can be found by applying two methods - the factoring method and the quadratic formula.

  By the factor method

  To factor a quadratic equation, $ax\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; letter-spacing:\; inherit;\; color:\; black;">2</sup>}+bx+c=0$

  Step 1) - We need to take two numbers such that their sum/difference is $b$ and product is $a\mathrm{c.}$

  Step 2) - Write the middle term as the sum or difference of the two numbers.

  Step 3) - Factor the first two terms and last two terms.

  Step 4) - Now both the terms should have a common factor. Take that common factor out.

  So, this complete process is the factor method.

  Hence, the quadratic equation becomes $(x+3)(x+1)=0.$

4. Linear Equation

• $A\; linear\; equation\; comprises\; an\; unknown\; variable\; and\; a\; no\; exponent\; more\; than\; 1.$

• $It\; uses\; concepts\; like\; Linear\; Equations\; with\; two\; unknowns\; and\; a\; Number\; of\; Solutions. <br\; style="word-break:\; break-word;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; letter-spacing:\; inherit;\; color:\; black;">$

• Example: If 5x + 2 = 3x + 18, what is the value of x?
  a) 4
  b) 6
  c) 8
  d) 10
  e) 12

  Sol: Start with the equation:
  5x + 2 = 3x + 18
  Move all terms involving x to one side by subtracting 3x from both sides:
  5x - 3x + 2 = 18
  2x + 2 = 18
  Move the constant term to the other side by subtracting 2 from both sides:
  2x = 16
  Divide both sides by 2 to solve for x:
  x = 8

Algebraic Formulas

There are three important algebra formulas that you need to mug up to solve the GMAT algebra questions. The formulas are given below -

• The difference in Two Squares: a²–b²=(a−b)(a+b)
• Squaring a Binomial: (a±b)²=a²±2ab+b²
• The Discriminant: D=b²–4ac

Example: If x² - 25 = 0, what is the value of x?

a) 5

b) -5

c) ±5

d) 25

e) 0

Sol: Recognize that x² - 25 is a difference of squares, which can be factored as:

x² - 25 = (x - 5)(x + 5) = 0

Set each factor equal to zero:

x - 5 = 0 → x = 5

x + 5 = 0 → x = -5

So, x = ±5.