2.7Foundation tier1MA1 · Section 2: Equations, formulae and identities

Quadratic expressions and equations

Key learnings

Rearrange to $=0$ , then pick a method
Every quadratic must first be written as $ax^2+bx+c=0$ . Then factorise if it factorises cleanly, use the quadratic formula $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ for decimal answers, or complete the square for an exact (surd) answer or the turning point.
Factors are not the same as solutions
Factorising gives $(x+p)(x+q)$ ; solving means setting each bracket to $0$ , so $x=-p$ or $x=-q$ . When a question says "hence solve", read the roots straight off the factors you already found — do not start again.
The discriminant $b^2-4ac$ counts the roots
If $b^2-4ac>0$ there are two real roots; if $b^2-4ac=0$ one repeated root; if $b^2-4ac<0$ no real roots. "Show it has no real solutions" is a discriminant argument, not a full solve.

Common mistakes

Drawn from real examiner reports.

Signs the wrong way round in the brackets

Choosing the correct factor pair but placing the signs wrongly, e.g. writing $(x+8)(x-3)$ for $x^2-5x-24$ instead of $(x-8)(x+3)$ . Expand back to check: $(x+8)(x-3)=x^2+5x-24$ , which is not $x^2-5x-24$ . Multiplying the brackets out again instantly catches a sign slip.

Re-factorising or using the formula after a "factorise" part

After factorising in part (i), students asked to "hence solve" try to factorise again or use the quadratic formula instead of reading the roots from their factors — and some give factors when solutions were asked for (or vice versa).

Completing the square with $a\ne 1$ — forgetting to multiply the constant

When $a\ne 1$ , the number produced inside the bracket must be multiplied by the leading coefficient. Students wrote $2x^2-24x+7 = 2(x-6)^2 - 36 + 7$ , leaving $-36$ when it should be $2\times(-36)=-72$ , i.e. $2(x-6)^2-72+7$ . For a negative leading coefficient, take out the $-1$ (or $-3$ ) first.

Dividing by $a$ in the formula, or not setting $=0$

In $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ the whole numerator is divided by $2a$ , not by $2$ . Watch the sign of $-b$ when $b$ is negative, keep the $\pm$ (plus or minus — two possibilities, one with each sign), and rearrange properly to $=0$ first.

Exam tips

Check factors by expanding, and reject invalid roots
Multiply your factors back out as a quick self-check — it catches sign errors at once. In a context (a length, a time) or a stated domain such as $x\ge 6$ , reject the root that cannot apply rather than quoting both, and do not round until the very end.

Revision notes

A quadratic equation must be written as $ax^2+bx+c=0$ before you solve it. A common slip is to leave a constant on the right (e.g. solving $x^2-x=3$ as if the right-hand side were $0$ ) — move everything to one side first.

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Quadratic expressions and equations

Key learnings

Common mistakes

Exam tips

Revision notes

First, rearrange to $=0$

Choosing a method

The discriminant $b^2-4ac$

Completing the square when $a\ne 1$ (a classic trap)

Don't mix these up — and always check

Flashcards