Quadratic Equations — chapter notes

A quadratic equation in $x$ is an equation that can be written as

where $a$, $b$, $c$ are real numbers. This is its standard form.

Always expand and bring all terms to one side before deciding: $(x + 2)^2 = x^2 + 5$ simplifies to $4x - 1 = 0$ (linear), while $x^3 - x^2 = x^3 + 4$ simplifies to $x^2 + 4 = 0$ (quadratic). Equations with $\dfrac1x$ are not quadratic as given, but may become quadratic after multiplying by $x$ ($x \ne 0$).

A real number $\alpha$ is a root (solution) of $ax^2 + bx + c = 0$ if $a\alpha^2 + b\alpha + c = 0$. The roots of the equation are the zeroes of the polynomial $ax^2 + bx + c$, i.e. the $x$-coordinates where the graph of $y = ax^2 + bx + c$ meets the $x$-axis. A quadratic equation has at most two roots.

To test a root, substitute: $2x^2 - 7x + 3 = 0$ has the root $3$ since $18 - 21 + 3 = 0$. If $1$ is a root, then $a + b + c = 0$. A known root gives an equation for an unknown coefficient.

1. Write the equation in standard form.
2. Split the middle term: find two numbers with product $ac$ and sum $b$.
3. Factorise in pairs: $(px + q)(rx + s) = 0$.
4. Zero-product rule: $px + q = 0$ or $rx + s = 0$.

Example: $6x^2 - x - 2 = 0$: $ac = -12 = (-4)(3)$; $(2x + 1)(3x - 2) = 0$; roots $-\dfrac12, \dfrac23$.

Cautions: never divide by $x$ ($x^2 = 9x$ has roots $0$ and $9$); the zero-product rule needs the right side to be zero — $(x - 2)(x + 3) = 6$ must first become $x^2 + x - 12 = 0$.

For $ax^2 + bx + c = 0$ with $a \ne 0$ and $b^2 - 4ac \ge 0$:

It works for every quadratic, including those that do not factorise. Read $a$, $b$, $c$ (with signs) from the standard form. Example: $2x^2 - 4x - 3 = 0$: $b^2 - 4ac = 40$, $x = \dfrac{4 \pm 2\sqrt{10}}{4} = \dfrac{2 \pm \sqrt{10}}{2}$.

Check with the polynomial relations: sum of roots $= -\dfrac ba$, product $= \dfrac ca$.

Not in the 2026–27 syllabus: solution (and derivation of the formula) by completing the square.

The discriminant of $ax^2 + bx + c = 0$ is $D = b^2 - 4ac$.

• $D > 0$: two distinct real roots (rational if $D$ is a perfect square and $a, b, c$ are integers).
• $D = 0$: two equal real roots, each $-\dfrac{b}{2a}$.
• $D < 0$: no real roots.

Real roots need $D \ge 0$. For equal roots, solve $D = 0$ for the unknown; expect two values when $k$ appears squared ($k^2 = 144 \Rightarrow k = \pm 12$), and reject any value that makes $a = 0$. If $a$ and $c$ have opposite signs, $D > 0$ always.

Steps: choose the unknown (with units) → express the other quantities → form the equation in standard form → solve → reject inadmissible roots (negative lengths, speeds, ages, counts; non-natural numbers; values excluded by the problem) → answer with units → check in the story.

Typical set-ups:

• Area: $x(x + 5) = 84$ gives $7$ m by $12$ m.
• Numbers: $n(n + 2) = 168$ gives $12$ and $14$.
• Speed–time: $\dfrac{180}{x} + \dfrac{180}{x + 15} = 7$ gives $45$ km/h.
• Fencing: both roots may be positive — use the other conditions (e.g. wall length) to decide.
• Possibility: if the equation has $D < 0$, the situation cannot happen (e.g. $70$ m of fence cannot enclose $320$ m²).