What a quadratic is
A quadratic is any expression where the highest power of the variable is 2, nothing higher, nothing lower will do. In its tidiest "standard form" it looks like this:
The letters \(a\), \(b\) and \(c\) are just numbers (the coefficients). The one rule that can't be broken is \(a \neq 0\): if \(a\) were zero, the \(x^2\) term would vanish and you'd be left with a plain straight-line equation, not a quadratic at all.
- Quadratic equation vs. quadratic function
- An equation like \(x^2 - 5x + 6 = 0\) asks "for what \(x\) is this true?" A function like \(f(x) = x^2 - 5x + 6\) is a machine: feed in any \(x\), get out a \(y\). Solving the equation is the same as asking where the function's graph crosses the \(x\)-axis.
Where this is assessed
Quadratics show up most in Criterion A (selecting and applying the right method) and Criterion C (communicating your working with correct notation). Word problems push into Criterion D: applying maths to a real situation and interpreting the answer.
The three forms of a quadratic
The same quadratic can be written three ways, and each one hands you a different fact for free. Fluency at MYP level is really about spotting which form makes your job easiest.
| Form | Looks like | What it hands you free |
|---|---|---|
| Standard | \(ax^2 + bx + c\) | The \(y\)-intercept, \(c\); the shape via \(a\). |
| Factored | \(a(x - p)(x - q)\) | The roots \(x = p\) and \(x = q\) at a glance. |
| Vertex | \(a(x - h)^2 + k\) | The turning point \((h,\,k)\) directly. |
Read that table again before an exam. Half of "I don't know where to start" moments disappear once you ask: which form is this, and which form do I want?
Solving by factorising
Factorising is the fastest route when the numbers are friendly. The goal is to rewrite the quadratic as two brackets multiplied together, then use the null factor law: if two things multiply to give zero, at least one of them must be zero.
Solve \(x^2 - 5x + 6 = 0\).
Quick check
Always test one root by substituting it back: \(2^2 - 5(2) + 6 = 4 - 10 + 6 = 0\). ✓ Costs ten seconds, catches most sign slips.
The quadratic formula
When factorising won't cooperate (awkward decimals, or roots that aren't whole numbers), the quadratic formula always works. It's the reliable workhorse, and it's worth knowing by heart:
The \(\pm\) is why a quadratic usually has two solutions: one from the \(+\) branch, one from the \(-\).
Solve \(2x^2 + 3x - 5 = 0\), giving exact values.
The classic trap
\(-4ac\) with a negative \(c\) becomes plus. Here \(-4(2)(-5) = +40\), not \(-40\). Losing that sign is the single most common quadratic-formula error, so slow down on it.
Completing the square
This method rewrites a quadratic in vertex form, which is how you find the turning point without a graph. It feels fiddly at first, then becomes routine.
Write \(x^2 + 6x + 5\) in the form \((x+h)^2 + k\).
The discriminant: how many solutions?
You can tell how many real solutions a quadratic has before solving it, just from the part under the square root. That expression, \(b^2 - 4ac\), is called the discriminant (symbol \(\Delta\)).
| Discriminant | What it means | Graph |
|---|---|---|
| \(b^2 - 4ac > 0\) | Two distinct real roots | Crosses the \(x\)-axis twice |
| \(b^2 - 4ac = 0\) | One repeated root | Just touches the \(x\)-axis |
| \(b^2 - 4ac < 0\) | No real roots | Never meets the \(x\)-axis |
This is a favourite exam question because it tests understanding, not grinding: "Show that \(x^2 + 2x + 5 = 0\) has no real solutions." You'd compute \(\Delta = 2^2 - 4(1)(5) = 4 - 20 = -16 < 0\), and that's the whole answer.
Reading the parabola
Every quadratic function graphs as a parabola, a smooth, symmetric U-shape. Four features are worth naming, because questions ask for them by name:
- Direction: if \(a > 0\) the U opens upward (a minimum); if \(a < 0\) it opens downward (a maximum). Remember it as "positive = happy, negative = sad."
- Roots (zeros): where it crosses the \(x\)-axis; these are the solutions of \(ax^2+bx+c=0\).
- \(y\)-intercept: where it crosses the \(y\)-axis, always at \((0,\,c)\).
- Vertex: the turning point, sitting exactly halfway between the roots on the axis of symmetry, \(x = -\dfrac{b}{2a}\).
Sketch, don't plot
In an exam you rarely need an accurate plot. Mark the two roots, the \(y\)-intercept, and the vertex, then join them with a smooth curve. Four points and the right direction is a full-mark sketch.
A real-world use: Criterion D
Quadratics model anything that rises and falls: a thrown ball, a jet of water, profit against price. Criterion D asks you to move between the maths and the situation and interpret, not just calculate.
A ball's height in metres after \(t\) seconds is \(h(t) = -5t^2 + 20t\). When does it hit the ground, and how high does it get?
The interpretation sentence at the end (turning "\(t=4\)" back into "lands after 4 seconds") is exactly what earns the Criterion D marks. Never leave your answer as a bare number in a word problem.
Check yourself
Try these before peeking. Working matters as much as the answer.
1. Solve \(x^2 - 7x + 12 = 0\) by factorising. +
Two numbers multiplying to \(12\) and adding to \(-7\): that's \(-3\) and \(-4\). So \((x-3)(x-4)=0\), giving \(x = 3\) or \(x = 4\).
2. How many real roots does \(x^2 + 4x + 4 = 0\) have? +
Discriminant \(= 4^2 - 4(1)(4) = 16 - 16 = 0\). A zero discriminant means exactly one repeated root (the parabola just touches the \(x\)-axis). For the record, it's \(x = -2\).
3. Use the quadratic formula to solve \(x^2 - 4x + 1 = 0\), leaving your answer in surd form. +
\(a=1,\,b=-4,\,c=1\). Then \(x = \dfrac{4 \pm \sqrt{16 - 4}}{2} = \dfrac{4 \pm \sqrt{12}}{2} = \dfrac{4 \pm 2\sqrt{3}}{2}\), so \(x = 2 \pm \sqrt{3}\).
Exam-day summary
If you remember nothing else, remember the decision tree:
- Can it factorise easily? Do that: it's fastest.
- Not obvious? Reach for the quadratic formula; it never fails.
- Asked for the turning point or a max/min? Complete the square, or use \(x = -\tfrac{b}{2a}\).
- Asked only "how many solutions"? Just the discriminant: no need to solve.
- Word problem? Solve the maths, then write one sentence translating the answer back into the situation.
Marks come from method
MYP examiners reward clear, followable working (Criterion C) even when the final number slips. Write each line so a reader could redo your steps without guessing, and that habit alone lifts grades.
This is a sample note showing the format for every topic in the library. Spotted an error or want a topic added? That's the kind of feedback that makes the notes better.