Elementary Algebra

🔷 Basic Terminology

Term	Description	Example
Variable	A symbol (usually x, y, z) representing a number	x, y
Constant	A fixed value	2, -5, 7.5
Coefficient	A number multiplying a variable	In 3x, 3 is the coefficient
Term	A single variable or number	5x, -3y, 2
Expression	A combination of terms	3x + 2y - 5
Equation	Two expressions are set equal	3x + 5 = 11

🔷 Types of Algebraic Expressions

• Monomial: One term → $5x$

• Binomial: Two terms → $x + 3$

• Trinomial: Three terms → $x^2 + 2x + 1$

• Polynomial: More than one term, can be of any degree.

🔷 Operations on Algebraic Expressions

➤ Addition & Subtraction

• Combine like terms:

  $$4x + 3x - 2x = 5x$$

➤ Multiplication

• Use distributive law:

  $$(a + b)(c + d) = ac + ad + bc + bd$$

➤ Division

• Simplify by cancelling common factors:

  $$\frac{x^2-9}{x-3}=x+3$$

🔷 Important Algebraic Identities

These are must-know shortcuts:

Identity No.	Formula
(1)	$(a + b)^2 = a^2 + 2ab + b^2$
(2)	$(a - b)^2 = a^2 - 2ab + b^2$
(3)	$(a + b)(a - b) = a^2 - b^2$
(4)	$(x + a)(x + b) = x^2 + (a + b)x + ab$
(5)	$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$
(6)	$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
(7)	$a^3-b^3=(a-b)(a^2+ab+b^2)$

🔷 Linear Equations in One Variable

General Form:

$$ax+b=0\Rightarrow x=\frac{-b}{\mathrm{a}}$$

Example:

$$3x+5=14\Rightarrow x=\frac{9}{3}=3$$

🔷 Linear Equations in Two Variables

Form:

$$ax + by = c$$

Solve using:

• Substitution method

• Elimination method

• Cross multiplication

🔷 Quadratic Equations

General Form:

$$ax^2 + bx + c = 0$$

Solutions:

• Factorization Method

• Quadratic Formula:

  $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2\mathrm{a}}$$

Discriminant:

• If $D=b^2-4ac>\mathrm{0\; \to \; Real\; and\; distinct\; roots}$

• If $D = 0$ → Real and equal roots

• If $D<\mathrm{0\; \to \; Imaginary\; roots}$

🔷 Inequalities

Rule	Example
Add/Subtract the same value on both sides	$x + 5 < 10 \Rightarrow x < 5$
Multiply/Divide by a positive number: sign remains	$2x < 6 \Rightarrow x < 3$
Multiply/Divide by a negative number: reverse the sign	$-2x < 6 \Rightarrow x > -3$

🔷 Exponents and Powers

Rule	Expression	Result
Product of powers	$a^m \cdot a^n$	$a^{m+n}$
Power of a power	$(a^m)^n$	$a^{mn}$
Quotient of powers	$\frac{a^m}{a^n}$	$a^{m-n}$
Negative exponent	$a^{-n}$	$\frac{1}{a^n}$
Zero exponent	$a^0$	$1$

🔷 Algebraic Fractions

• Simplify by factoring:

  $$\frac{x^2-4}{x+2}=\frac{(x-2)(x+2)}{x+2}=x-2$$

🔷 Word Problems on Algebra

Types:

• Age Problems
• Work and Time
• Speed, Time & Distance
• Mixtures
• Number Problems
• Ratio and Proportion

Key Tip:

• Define variables clearly
• Set up the equation based on conditions
• Solve and check for logic

🔷 Common Shortcuts & Tricks

Problem Type	Shortcut
Product of 2 conjugates	$(a + b)(a - b) = a^2 - b^2$
Sum of the first n natural numbers	$\frac{n(n+1)}{2}$
Sum of squares	$\frac{n(n+1)(2n+1)}{6}$
Sum of cubes	${\left[\frac{n(n+1)}{2}\right]}^{\mathrm{2-}}$

Here are 5 multiple-choice questions (MCQs) based on Elementary Algebra, complete with correct answers and short, descriptive analyses for each:

🟦 Question 1: Simplifying Expressions

Q1. What is the simplified form of the expression:
(2x + 3)(x − 4)?

A) 2x² − 5x − 12
B) 2x² − 8x + 3
C) 2x² − 5x − 15
D) 2x² − 8x − 12

Correct Answer: A) 2x² − 5x − 12

🧠 Explanation:
Use the distributive (FOIL) method:
(2x)(x) = 2x²
(2x)(−4) = −8x
(3)(x) = 3x
(3)(−4) = −12
Now combine: 2x² − 8x + 3x − 12 = 2x² − 5x − 12

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🟦 Question 2: Identity Application

Q2. Which identity is used to expand (a + b)²?

A) a² − 2ab + b²
B) a² + 2ab + b²
C) a² − b²
D) a³ + b³

Correct Answer: B) a² + 2ab + b²

🧠 Explanation:
This is a standard algebraic identity:
(a + b)² = a² + 2ab + b²
This helps in quickly expanding binomials without full multiplication.

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🟦 Question 3: Solving Linear Equations

Q3. Solve: 5x − 3 = 2x + 6

A) x = 1
B) x = 3
C) x = 9
D) x = −3

Correct Answer: B) x = 3

🧠 Explanation:
Step 1: Bring variables to one side
5x − 2x = 6 + 3
3x = 9
x = 9 ÷ 3 = 3

—————

🟦 Question 4: Quadratic Factorization

Q4. Factor the expression: x² − 7x + 12

A) (x − 3)(x − 4)
B) (x − 6)(x − 1)
C) (x − 2)(x − 5)
D) (x − 3)(x + 4)

Correct Answer: A) (x − 3)(x − 4)

🧠 Explanation:
We look for two numbers whose sum is −7 and product is +12.
−3 and −4 work: (−3) + (−4) = −7 and (−3)(−4) = 12
So the factorization is (x − 3)(x − 4)

—————

🟦 Question 5: Exponent Rule

Q5. Simplify: (2x³)²

A) 4x⁵
B) 4x⁶
C) 2x⁶
D) 8x⁵

Correct Answer: B) 4x⁶

🧠 Explanation:
Use (ab)^n = a^n × b^n
(2x³)² = 2² × (x³)² = 4 × x⁶ = 4x⁶