Solving quadratic equations

The standard form

• A quadratic equation has a squared term and nothing higher:

$$ax^2 + bx + c = 0 \qquad (a \neq 0)$$

• Always rearrange to this form first, with everything on one side and zero on the other. None of the methods work otherwise.
• A quadratic has at most two real solutions, also called roots.

Method 1 — factorising

• The fastest method when it works.

  • Rearrange to $= 0$.
  • Factorise the left-hand side.
  • Set each factor to zero and solve.

• This works because of the null factor law: if $AB = 0$ then $A = 0$ or $B = 0$.

• Worked through — solve $x^2 + 2x = 15$:

  • Rearrange: $x^2 + 2x - 15 = 0$
  • Factorise: $(x+5)(x-3) = 0$
  • Solve: $x = -5$ or $x = 3$

• Do not divide by $x$. Solving $x^2 = 5x$ by dividing gives $x = 5$ and loses the solution $x = 0$. Factorise instead: $x(x-5) = 0$ gives both.

Method 2 — the quadratic formula

• Works every time, whether or not the quadratic factorises:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

• Each variable:

  • $a$ — the coefficient of $x^2$
  • $b$ — the coefficient of $x$
  • $c$ — the constant term

• Include the signs when you read off $a$, $b$ and $c$. For $2x^2 - 7x - 4 = 0$, $b = -7$ and $c = -4$.

• Worked through for $2x^2 - 7x - 4 = 0$:

  • $$x = \dfrac{7 \pm \sqrt{49 + 32}}{4} = \dfrac{7 \pm 9}{4}$$
  • $x = 4$ or $x = -\dfrac{1}{2}$

• The $\pm$ gives two answers. Giving only one is a half-answer.

Method 3 — completing the square

• Useful when the question asks for the vertex, or for an exact answer in surd form.

  • Rewrite as $(x + p)^2 = q$.
  • Square root both sides, remembering $\pm$.
  • Solve for $x$.

• Worked through — solve $x^2 - 6x + 2 = 0$:

  • $(x-3)^2 - 9 + 2 = 0$, so $(x-3)^2 = 7$
  • $$x - 3 = \pm\sqrt{7}$$
  • $$x = 3 \pm \sqrt{7}$$

• The $\pm$ appears when the square root is taken, and it must not be dropped.

Rational (fractional) roots

• The 2026 specification requires candidates to "understand the meaning of rational (fractional) numbers in regard to the roots of equations".
• A root such as $x = \frac{3}{2}$ comes from a factor $(2x - 3)$, because $2x - 3 = 0$ gives $x = \frac{3}{2}$.
• Working backwards: if a quadratic has roots $\frac{1}{2}$ and $-4$, its factors are $(2x - 1)$ and $(x + 4)$, so the equation is $2x^2 + 7x - 4 = 0$.
• Note that a fractional root always signals a leading coefficient greater than 1.

Building a quadratic from its roots

• If the roots are $\alpha$ and $\beta$:

$$x^2 - (\alpha + \beta)x + \alpha\beta = 0$$

• Sum of roots $= -\dfrac{b}{a}$ and product of roots $= \dfrac{c}{a}$.
• The 2025 report lists "formed a quadratic equation from the solution" as an Achieved behaviour.

Which method to choose

• Factorising — try it first; it is quickest when the numbers are friendly.
• The formula — use when factorising fails, or when the coefficients are awkward.
• Completing the square — use when the question asks for the vertex, an exact surd answer, or the maximum/minimum of the quadratic.