Introduction to Quadratic Equations
Introduction and definition of quadratic equations.
📐 What is a Quadratic Equation?
In mathematics, an equation involves variables (like x or y) set equal to zero or to another expression. When the highest power of the variable in that equation is exactly 2, it is called a quadratic equation.
Any equation of the form p(x) = 0, where p(x) is a polynomial of degree 2, is called a quadratic equation. The word "quadratic" comes from the Latin quadratus, meaning "square," because the variable is raised to the power of 2 (squared).
The key condition is simple: the highest degree (exponent) of the variable must be 2. Equations with degree 1 are linear, degree 3 are cubic — only degree 2 are quadratic.
✏️ Recognising Quadratic Equations — Examples
The following equations are all quadratic because the largest power of the variable in each is 2:
Degree = 2
Degree = 2
Degree = 2
Notice that the equations above are not yet in the neat, organised form. The terms can appear in any order. To work with them properly, we rearrange them into standard form.
📊 Standard Form of a Quadratic Equation
Writing the terms of p(x) in descending order of their degrees (degree 2 first, then degree 1, then the constant) gives us the standard form:
ax² + bx + c = 0
where a ≠ 0, and a, b, c are real numbers called coefficients
The condition a ≠ 0 is crucial — if a were 0, the x² term would vanish and the equation would no longer be quadratic (it would become linear).
🔢 Identifying the Coefficients a, b, and c
Once an equation is in standard form, you can directly read off the three coefficients. Study the table below carefully — this is a very common exam question:
| Equation (Standard Form) | a (x² coeff.) | b (x coeff.) | c (constant) |
|---|---|---|---|
| 2x² − 5x + 3 = 0 | 2 | −5 | 3 |
| 5y² − 1 = 0 | 5 | 0 | −1 |
| x² − x = 0 | 1 | −1 | 0 |
| √2·x² − ⅓ = 0 | √2 | 0 | −⅓ |
Tip: When a term is missing, its coefficient is 0. For example, in 5y² − 1 = 0 there is no y term, so b = 0. Similarly in x² − x = 0 there is no constant, so c = 0. Never leave b or c undefined — always write 0!
🔍 Quick Check: Is It Quadratic?
Use this simple decision process whenever you need to check whether an equation is quadratic. The only thing you need to find is the degree of the equation — the highest power of the variable.
🧠 Worked Problems — Check if Quadratic
Let's go through each problem from the "Try This" section of the lesson step-by-step:
✏️ Solved Examples
x⁴ + 1 = 2x²
→ x⁴ − 2x² + 1 = 0
The highest degree is now 4. Even though the original looks quadratic, multiplying to clear the denominator raises the degree!
Left: 6x² + 5x + 1
Right: bx² − 3bx + 2b
Bring all terms to one side:
(6 − b)x² + (5 + 3b)x + (1 − 2b) = 0
For this to be quadratic, the coefficient of x² must be non-zero: 6 − b ≠ 0, i.e., b ≠ 6.
🌍 Where Are Quadratic Equations Used in Real Life?
Quadratic equations are not just abstract maths — they appear all around us in real-world situations:
⚠️ Common Mistakes to Avoid
Forgetting that a ≠ 0. If the coefficient of x² is 0, the equation is not quadratic — always check!
Not simplifying before checking degree. Like problem (iv): x² + 1/x² = 2 looks quadratic but becomes degree 4 after simplification.
Writing b or c as "not present" instead of 0. Always write b = 0 or c = 0 explicitly when those terms are missing.
Assuming every equation with x² in it is automatically quadratic. Check whether, after simplification, x² terms cancel each other out.
- Standard form is mandatory — always rearrange before identifying a, b, and c.
- Degree, not the number of terms, determines if an equation is quadratic.
- b and c can be zero, but a must never be zero in a quadratic equation.
- Expand and simplify before concluding — products of two linear terms may or may not give quadratics depending on coefficients.