An expression is quadratic when its highest power of the unknown is 2 after you simplify. Check the highest power only once all brackets are expanded and like terms are collected.
This is the first lesson in SPM Mathematics quadratic functions and equations. Every later skill, from solving to graphs, starts by knowing what a quadratic looks like.
What does the general form look like?
The general form is ax² + bx + c, where a, b and c are numbers and a is not 0. The letter a goes with x², b goes with x, and c is the constant.
Take 4x² − 7x + 2. Here a = 4, b = −7 and c = 2. The sign belongs to the coefficient, so b is −7 and not 7.
Four expressions, one rule
Here are four original expressions. Decide which are quadratic.
| Expression | After simplifying | Highest power | Quadratic? |
|---|---|---|---|
| 5x² − 3 | 5x² − 3 | 2 | Yes, with b = 0 |
| x(x + 6) | x² + 6x | 2 | Yes, with c = 0 |
| (x + 2)² − x² | 4x + 4 | 1 | No, it is linear |
| x³ − x(x² − 9) | 9x | 1 | No, it is linear |
The third and fourth rows are the traps. They look as if they hold squares or cubes, but the highest powers cancel. Only expanding shows the truth.
The mistake that costs marks
The usual slip is judging from the first glance. A student sees (x + 2)² − x² and decides “squared, so quadratic”, then tries to factorise and finds no two numbers that work.
The contrast is simple. The wrong method reads the expression as written, and the right method expands first:
(x + 2)² − x² = x² + 4x + 4 − x² = 4x + 4.
The x² terms cancel, so the equation 4x + 4 = 0 is linear and gives x = −1 directly.
Naming a, b and c when the expression is not tidy
Sometimes the terms are out of order or split across both sides of an equals sign. Take 7 − 2x² = 3x. Move everything to one side first, keeping the x² term positive if you can.
Add 2x² to both sides and subtract 7: 0 = 2x² + 3x − 7. So a = 2, b = 3, c = −7.
You would get the same equation with every sign reversed, but then a = −2, b = −3 and c = 7. Both are valid, and most factorising is easier when a is positive.
Check yourself
Which of these is quadratic, and what are a, b and c for it? (A) 2x + 9 (B) (x − 1)(x + 5) − x² (C) (2x − 1)(x + 4) − 3x.
Answer
(A) is linear, the highest power is 1.
(B) expands to x² + 4x − 5 − x² = 4x − 5, which is linear.
(C) expands to 2x² + 8x − x − 4 − 3x = 2x² + 4x − 4, so it is quadratic with a = 2, b = 4, c = −4.
What to study next
Once you can name a, b and c, move to solving factorisable quadratic equations. You can also see how a changes the shape of the graph in the quadratic graph and roots explorer.
If expanding and simplifying still cost you marks, online one-to-one Mathematics tuition gives a teacher time to check that layer of working with you.