1. Introduction
A quadratic equation is an equation that can be written in the form:
\( ax^2 + bx + c = 0 \)
where a, b, and c are real numbers, and a ≠ 0. The term \(ax^2\) is called the quadratic term, \(bx\) is the linear term, and \(c\) is the constant term.
This form is known as the standard form of a quadratic equation. Every quadratic equation can be rearranged into this form.
2. Identifying a Quadratic Equation
A quadratic equation must:
- Contain the term \(x^2\), with coefficient ≠ 0
- Have the highest power of \(x\) equal to 2
- Possibly (but not necessarily) include \(x\) and a constant term
Examples of valid quadratic equations:
- \(2x^2 + 3x - 5 = 0\)
- \(x^2 - 9 = 0\)
- \(7x - x^2 + 4 = 0\)
Examples that are NOT quadratic equations:
- \(3x + 2 = 0\) (linear)
- \(x^3 - 4x = 0\) (cubic)
- \(5 = 0\) (no variable)
3. Converting an Equation to Standard Form
Sometimes quadratic equations are not written in standard form. To convert them, rewrite all terms on one side so that the equation equals 0, and combine like terms.
Steps:
- Move all terms to one side of the equation.
- Arrange in decreasing powers of \(x\): \(x^2\), then \(x\), then constant.
- Simplify coefficients if needed.
3.1. Examples
Example 1: Convert \(3x^2 = 5x - 2\) to standard form.
Move all terms to one side:
\(3x^2 - 5x + 2 = 0\)
Example 2: Convert \(7 = x - x^2\).
Rearrange:
\(-x^2 + x - 7 = 0\)
Multiply by -1 if preferred:
\(x^2 - x + 7 = 0\)
4. Recognising Coefficients a, b, and c
In the standard form \(ax^2 + bx + c = 0\):
- \(a\) = coefficient of \(x^2\)
- \(b\) = coefficient of \(x\)
- \(c\) = constant term
4.1. Examples
| Equation | a | b | c |
|---|---|---|---|
| \(2x^2 + 3x - 5 = 0\) | 2 | 3 | -5 |
| \(x^2 - 9 = 0\) | 1 | 0 | -9 |
| \(5x - x^2 + 4 = 0\) | -1 | 5 | 4 |
5. Special Forms That Still Represent Quadratic Equations
Some equations might not look quadratic at first, but can be rewritten into the standard form:
- \(x(x - 5) = 6\) → expand → \(x^2 - 5x - 6 = 0\)
- \((x - 3)(x + 2) = 4\) → expand → \(x^2 - x - 6 = 4\) → rearrange → \(x^2 - x - 10 = 0\)
- \(\dfrac{1}{x} + x = 2\) → multiply by x → \(1 + x^2 = 2x\) → \(x^2 - 2x + 1 = 0\)
6. Common Mistakes
- Forgetting to move all terms to one side.
- Incorrect signs during rearrangement.
- Assuming any equation containing \(x^2\) is automatically quadratic (not true if the equation simplifies differently).
- Not combining like terms carefully.
7. Quick Practice
Convert the following to standard form:
- \(x - 4 = 2x^2\)
- \(3 = x(x + 5)\)
- \(5x - 7 = x^2\)
- \(4x(x - 1) = 3x + 2\)
8. Summary
- The standard form of a quadratic equation is \(ax^2 + bx + c = 0\).
- A quadratic must have \(a ≠ 0\).
- Any equation with highest power 2 can be rearranged into standard form.
- Always arrange terms in decreasing powers and simplify carefully.