Introduction to Equations
An equation is a fundamental mathematical statement that shows two expressions are equal, denoted by the equal sign (=). It consists of two main parts:
- Left-Hand Side (LHS)
- Right-Hand Side (RHS)
The key principle is that LHS = RHS, meaning both sides balance each other. Equations are essential in algebra, physics, engineering, and everyday problem-solving.
Key Components of an Equation
- Variables – Unknown values represented by letters (e.g., x, y).
- Constants – Fixed numbers (e.g., 3, 7).
- Operators – Mathematical symbols (+, −, ×, ÷).
- Equal Sign (=) – Indicates both sides have the same value.
Example of a Simple Equation
Consider the equation:
2𝑥 + 3 = 7
- LHS: 2𝑥 + 3
- RHS: 7
- Goal: Solve for *x* to make the equation true.
Types of Equations in Mathematics
Equations can be classified into different categories based on their structure:
1. Linear Equations
- Highest power of the variable is 1.
- Forms a straight line when graphed.
- Example: 3𝑥 − 5 = 10
2. Quadratic Equations
- Highest power is 2 (e.g., 𝑥²).
- Forms a parabola when graphed.
- Example: 𝑥² + 5𝑥 + 6 = 0
3. Polynomial Equations
- Include higher powers (e.g., 𝑥³, 𝑥⁴).
- Can have multiple solutions.
- Example: 2𝑥³ − 4𝑥² + 𝑥 = 0
4. Exponential Equations
- Variables appear in exponents.
- Used in growth/decay problems.
- Example: 2ˣ = 16
5. Trigonometric Equations
- Involve trigonometric functions (sin, cos, tan).
- Used in wave and circular motion problems.
- Example: sin(𝑥) = 0.5
6. Differential Equations
- Contain derivatives of functions.
- Used in physics and engineering.
- Example: dy/dx = 2x
How to Solve an Equation (Step-by-Step Guide)
Let’s solve the example 2𝑥 + 3 = 7:
- Isolate the variable term (2𝑥):
Subtract 3 from both sides:
2𝑥 = 7 − 3
2𝑥 = 4 - Solve for 𝑥:
Divide both sides by 2:
𝑥 = 4 ÷ 2
𝑥 = 2 - Verify the solution:
Substitute 𝑥 = 2 back into the original equation:
2(2) + 3 = 7 → 4 + 3 = 7 (Correct!)
Quadratic Equations: The Ultimate Guide with Examples & Practice Problems
1. What is a Quadratic Equation?
A quadratic equation is any equation that can be written in the standard form:ax2+bx+c=0ax2+bx+c=0
Where:
- a, b, c are real numbers (a ≠ 0)
- x is the variable
- ax² is the quadratic term (must be present)
- bx is the linear term
- c is the constant term
Example of Quadratic Equations
2×2+5x−3=0(Standard Form)x2−9=0(Missing linear term)3×2+4x=0(Missing constant term)2x2+5x−3=0(Standard Form)x2−9=0(Missing linear term)3x2+4x=0(Missing constant term)
2. 3 Methods to Solve Quadratic Equations
There are three primary methods to solve quadratics. Let’s explore each with examples.
Method 1: Factorization (Splitting the Middle Term)
Best for: Equations that can be easily factored
Steps:
1️⃣ Find two numbers that:
- Multiply to a × c
- Add to b
2️⃣ Split the middle term using these numbers
3️⃣ Factor by grouping
4️⃣ Set each factor = 0 and solve
Example: Solve x2+5x+6=0x2+5x+6=0
- a = 1, b = 5, c = 6
- Find two numbers: 2 and 3 (since 2×3=62×3=6 and 2+3=52+3=5)
- Rewrite:x2+2x+3x+6=0x(x+2)+3(x+2)=0(x+2)(x+3)=0x2+2x+3x+6=0x(x+2)+3(x+2)=0(x+2)(x+3)=0
- Solutions: x = -2 or x = -3
Method 2: Completing the Square
Best for: Equations where factoring is difficult
Steps:
1️⃣ Move the constant term to the other side
2️⃣ Add (b2)2(2b)2 to both sides
3️⃣ Rewrite as a perfect square
4️⃣ Take the square root of both sides
5️⃣ Solve for x
Example: Solve x2+6x−16=0x2+6x−16=0
- Move constant:x2+6x=16x2+6x=16
- Add (62)2=9(26)2=9:x2+6x+9=25x2+6x+9=25
- Rewrite as square:(x+3)2=25(x+3)2=25
- Take square root:x+3=±5x+3=±5
- Solutions: x = 2 or x = -8
Method 3: Quadratic Formula (Most Reliable)
Works for ALL quadratic equations
Formula:x=−b±b2−4ac2ax=2a−b±b2−4ac
Steps:
1️⃣ Identify a, b, c
2️⃣ Compute the discriminant (D = b² – 4ac)
3️⃣ Plug into the formula
4️⃣ Simplify
Example: Solve 2×2+3x−2=02x2+3x−2=0
- a = 2, b = 3, c = -2
- Discriminant:D=32−4(2)(−2)=9+16=25D=32−4(2)(−2)=9+16=25
- Apply formula:x=−3±254=−3±54x=4−3±25=4−3±5
- Solutions: x = 0.5 or x = -2
3. Practice Problems (With Solutions)
Test your skills with these 5 quadratic equations:
1️⃣ x2−7x+12=0x2−7x+12=0
Solution: x = 3 or x = 4
2️⃣ 3×2+5x−2=03x2+5x−2=0
Solution: x = -2 or x = 1/3
3️⃣ 2×2−4x−6=02x2−4x−6=0
Solution: x = 3 or x = -1
4️⃣ x2+8x+15=0x2+8x+15=0
Solution: x = -3 or x = -5
5️⃣ 5×2+6x−8=05x2+6x−8=0
Solution: x = 0.8 or x = -2
4. Multiple-Choice Questions (MCQs)
Test your knowledge with these 5 MCQs:
Q1. The roots of x2−5x+6=0x2−5x+6=0 are:
A) 2, 3
B) -2, -3
C) 1, 6
✅ Answer: A
Q2. The discriminant of ax2+bx+c=0ax2+bx+c=0 is:
A) b2−4acb2−4ac
B) b2+4acb2+4ac
✅ Answer: A
Q3. If discriminant D<0D<0, the roots are:
A) Real and equal
B) Imaginary
✅ Answer: B
Q4. The solution to x2+4x+4=0x2+4x+4=0 is:
A) x = -2 (double root)
B) x = 2
✅ Answer: A
Q5. In 3x2−7x+2=03x2−7x+2=0, the value of b is:
A) -7
B) 3
✅ Answer: A
5. Quick Method Selection Guide
| Method | Best When… |
|---|---|
| Factorization | Equation easily factorable |
| Completing Square | Leading coefficient = 1 |
| Quadratic Formula | Always works (most reliable) |
Pro Tip: Use the Quadratic Formula if unsure—it works every time!
Understanding Mathematical Expressions: A Complete Guide with Examples & MCQs
What is a Mathematical Expression?
A mathematical expression is a combination of numbers, variables (like x, y), and operations (+, −, ×, ÷) that represents a value.
🔹 Key Feature: It does not contain an equal sign (=).
🔹 If it has “=” → It becomes an equation.
Types of Mathematical Expressions (With Examples)
1. Numeric Expressions (Numbers Only)
- Example:
5 + 3 - Example:
12 ÷ 4 × 2
2. Algebraic Expressions (Variables Involved)
- Simple:
x − 4 - Complex:
2a + 3b − 7
3. Polynomial Expressions (Higher Powers)
- Example:
x² + 3x − 1
Expressions vs. Equations: Key Differences
| Feature | Expression | Equation |
|---|---|---|
| Equal Sign? | ❌ No | ✔️ Yes |
| Example | 5 + 3 | 5 + 3 = 8 |
| Purpose | Represents a value | Shows equality |
5 Multiple-Choice Questions (MCQs) with Solutions
Q1. Which of the following is an expression?
A) 5 + 3 = 8
B) 7 × 2
C) x = 10
D) y + 4 = 12
✅ Answer: B
📌 Explanation: Only 7 × 2 has no equal sign.
Q2. Identify the expression:
A) a − 3
B) 2x = 4
C) 5 = 5
D) x + 3 = 10
✅ Answer: A
📌 Explanation: a − 3 has no equality sign.
Q3. Which one is NOT an expression?
A) 4y + 7
B) 2a − 5b
C) 5 + 2 = 7
D) x² + 3x − 1
✅ Answer: C
📌 Explanation: 5 + 2 = 7 is an equation.
Q4. Evaluate 3x + 5 if x = 2.
A) 8
B) 9
C) 11
D) 15
✅ Answer: C
📌 Solution: 3(2) + 5 = 6 + 5 = 11.
Q5. Which expression has two variables?
A) 4a + 7b
B) 5m
C) y − 2
D) 10p
✅ Answer: A
📌 Explanation: 4a + 7b contains a and b.
Bonus: 3 Advanced-Level MCQs (Challenge Yourself!)
Q6. Simplify: 2(x + 3) − 4x
A) −2x + 6
B) 6x − 4
C) 2x + 6
D) −4x + 2
✅ Answer: A
📌 Solution: 2x + 6 − 4x = −2x + 6.
Q7. What is the degree of 3x²y + 4xy − 7?
A) 1
B) 2
C) 3
D) 4
✅ Answer: C
📌 Explanation: The highest power sum (x²y → 2+1=3).
Q8. If a = 3 and b = −1, find 2a² − b³.
A) 17
B) 19
C) 21
D) 23
✅ Answer: B
📌 Solution: 2(3)² − (−1)³ = 18 − (−1) = 19.
Advanced Equations: 10 Challenging MCQs with Step-by-Step Solutions
