Quadratic Equations • Topic 1 of 3

Standard Form of a Quadratic Equation

What is the standard form of a quadratic equation? A quadratic equation is an equation of the form \(ax^{2} + bx + c\) = 0, where a, b, and c are real numbers, and a ≠ 0. The highest power of the variable x is 2 (hence the name "quadratic", from "quadratus" meaning square).

Why must a ≠ 0? If a = 0, the equation becomes bx + c = 0, which is linear, not quadratic.

Examples of quadratic equations:

  • \(x^{2} + 5x + 6\) = 0 (a=1, b=5, c=6)
  • \(2x^{2} - 4x\) = 0 (a=2, b=−4, c=0)
  • \(x^{2} - 9\) = 0 (a=1, b=0, c=−9)

Not quadratic:

  • \(x^{3} + 2x + 1\) = 0 (degree 3 — cubic)
  • 2x + 5 = 0 (degree 1 — linear)

Real-life analogy: Think of a quadratic equation as a "square-shaped" problem. If you throw a ball upward, its height over time follows a quadratic pattern — that's why quadratics are used to model projectiles, bridges, and even profit maximization!

┌─────────────────────────────────────────────────────────────┐
│         STANDARD FORM OF QUADRATIC EQUATION                  │
└─────────────────────────────────────────────────────────────┘

GENERAL STRUCTURE:

        ax²  +  bx  +  c  =  0
         │       │       │
         │       │       └── Constant term
         │       └── Coefficient of x (linear term)
         └── Coefficient of x² (quadratic term)

CONDITION: a ≠ 0 (otherwise it becomes linear)


IDENTIFYING a, b, c FROM DIFFERENT FORMS:

┌─────────────────────────────┬──────┬──────┬──────┐
│        EQUATION             │  a   │  b   │  c   │
├─────────────────────────────┼──────┼──────┼──────┤
│   3x² + 5x - 2 = 0          │  3   │  5   │  -2  │
│   x² - 7x = 0               │  1   │  -7  │  0   │
│   4x² - 9 = 0               │  4   │  0   │  -9  │
│   -2x² + x + 3 = 0          │  -2  │  1   │  3   │
│   x² = 4                    │  1   │  0   │  -4  │
└─────────────────────────────┴──────┴──────┴──────┘


CONVERTING TO STANDARD FORM:

     Equation: (x + 3)(x - 2) = 0
           │
           ▼ Expand
     x² - 2x + 3x - 6 = 0
           │
           ▼ Combine like terms
     x² + x - 6 = 0  ✓ (a=1, b=1, c=-6)

     
     Equation: 2x² + 5 = 3x
           │
           ▼ Bring all terms to LHS
     2x² + 5 - 3x = 0
           │
           ▼ Rearrange in descending order
     2x² - 3x + 5 = 0  ✓ (a=2, b=-3, c=5)


WHY QUADRATIC? REAL-LIFE SHAPE:

    Path of a thrown ball:
    
        height
          ↑
        10├─────╱╲─────
          │    ╱  ╲
        5├───╱    ╲───
          │  ╱      ╲
        0└─╱────────╲──→ time
          ╱          ╲
    
    This parabolic shape is described by a quadratic equation!
Solved Examples
1
Worked Example
Identify a, b, and c for the quadratic equation \(3x^{2} - 2x + 7\) = 0.
Solution
  1. Step 1: Compare with standard form \(ax^{2} + bx + c\) = 0
  2. Step 2: Coefficient of \(x^{2}\) is 3 → a = 3
  3. Step 3: Coefficient of x is −2 → b = −2
  4. Step 4: Constant term is 7 → c = 7

Answer: a = 3, b = −2, c = 7

2
Worked Example
Write the equation (x + 2)\(^{2}\) = 9 in standard quadratic form and identify a, b, c.
Solution
  1. Step 1: Expand LHS: (x + 2)\(^{2}\) = \(x^{2} + 4x + 4
  2. Step\) 2: Equation becomes: \(x^{2} + 4x + 4\) = 9
  3. Step 3: Bring all terms to LHS: \(x^{2} + 4x + 4 - 9\) = 0
  4. Step 4: Simplify: \(x^{2} + 4x - 5\) = 0
  5. Step 5: Compare: a = 1, b = 4, c = −5

Answer: \(x^{2} + 4x - 5\) = 0; a=1, b=4, c=−5

3
Worked Example
Determine the value of k for which the equation (k+1)\(x^{2} + 2\)(k+1)x + 2 = 0 becomes a linear equation.
Solution
  1. Step 1: For the equation to be linear, the coefficient of \(x^{2}\) must be zero
  2. Step 2: Set a = 0: k + 1 = 0
  3. Step 3: Solve: k = −1
  4. Step 4: Check: Substitute k = −1: (0)\(x^{2} + 2\)(0)x + 2 = 0 → 2 = 0 (false, so no solution actually)
  5. Step 5: Wait — if k = −1, the equation becomes 2 = 0 which has no solution. So no value makes it a valid linear equation. The question likely expects k = −1 makes it not quadratic, but it becomes inconsistent.

Answer: k = −1 (equation becomes inconsistent, not truly linear)

Key Points

  • Standard form: \(ax^{2} + bx + c\) = 0, where a ≠ 0
  • a, b, c are real numbers (coefficients)
  • a is the coefficient of \(x^{2}\), b of x, c is the constant
  • If a = 0, the equation is linear, not quadratic
  • Any quadratic equation can be rewritten in standard form
  • Degree of a quadratic equation is always 2
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.The standard form of a quadratic equation is:
Explanation: General quadratic form.
Q2.The degree of a quadratic equation is:
Explanation: Highest power $2$.
Q3.The roots of $(x-1)(x+2)=0$ are:
Explanation: Set each factor to $0$.
Q4.In $ax^2+bx+c=0$, we require:
Explanation: Otherwise it is not quadratic.
Practice more MCQs → 📝 Assignment · 35 marks