Quadratic Functions

Quadratics are the connective tissue of 0606: they appear in their own right and then resurface inside simultaneous equations, circle geometry, trig equations and calculus. The topic’s three pillars, completing the square, the discriminant, and inequalities, are pure method, and each has a fixed exam routine.

Completing the square (and what it’s actually for)

Express $ax^2 + bx + c$ in the form $a(x + p)^2 + q$:

$2x^2 - 12x + 23 = 2(x^2 - 6x) + 23 = 2[(x - 3)^2 - 9] + 23 =$ $2(x - 3)^2 + 5$

Show the factor-out line and the half-the-coefficient step, both carry method credit. The form answers four exam questions at once:

• Vertex: $(3, 5)$; minimum value $5$ at $x = 3$ (maximum if $a < 0$)
• Range: $f(x) \ge 5$, the standard route to range questions
• Number of roots: $2(x - 3)^2 + 5 = 0$ has none, since the left side is always $\ge 5$
• Inverse functions: the restricted-domain setup runs through the vertex

“Express in the form” is a command phrase: the accuracy marks attach to the exact format requested.

The discriminant: the syllabus’s favourite tool

For $ax^2 + bx + c = 0$, the discriminant $b^2 - 4ac$ determines root nature: $> 0$ two distinct real roots; $= 0$ equal (repeated) roots; $< 0$ no real roots. 0606 rarely asks it bare, it dresses it up:

• “Find the values of $k$ for which $kx^2 + 4x + k = 0$ has equal roots” $\to$ set $b^2 - 4ac = 0 \to 16 - 4k^2 = 0 \to k = \pm 2$
• “Show the line $y = x + 3$ does not meet the curve $y = x^2 + 2x + 7$” $\to$ substitute, collect, show $b^2 - 4ac < 0$, conclude in words
• “Find $k$ so the line is a tangent to the curve” $\to$ substitute and set $b^2 - 4ac = 0$

The routine never changes: substitute $\to$ rearrange to $= 0 \to$ write the line "$b^2 - 4ac \ldots$" explicitly $\to$ solve/conclude. The written discriminant statement is an M mark; the verbal conclusion is frequently the final A/B mark.

Quadratic inequalities: sketch, always

To solve $x^2 - x - 12 > 0$: factorise $(x - 4)(x + 3) > 0$, find roots $x = 4$, $x = -3$, sketch the parabola, read where it’s above the axis: $x < -3$ or $x > 4$. Two habits protect the marks: the sketch (sign errors are nearly impossible with the picture in front of you), and writing two-region answers as two inequalities, never the impossible "$-3 > x > 4$". Inside-region answers ($< 0$ cases) come out as a single interval: $-3 < x < 4$. These inequalities also gatekeep discriminant range questions. “find $k$ such that … has real roots” ends in a quadratic inequality in $k$.

Line–curve intersection

Substitute the linear into the quadratic, collect everything on one side, and the resulting quadratic’s solutions are the intersection $x$-values, finish by finding the $y$-values from the linear equation (less algebra, fewer slips). Two points $=$ chord; one repeated point $=$ tangent; none $=$ miss. The full simultaneous treatment is in simultaneous equations.

Worked exam-style question

The line $y = 2x + k$ is a tangent to the curve $y = x^2 - 4x + 13$. Find $k$ and the point of contact.

Substitute: $x^2 - 4x + 13 = 2x + k \to x^2 - 6x + (13 - k) = 0$ (M, collected to zero) Tangent $\Rightarrow b^2 - 4ac = 0$: $36 - 4(13 - k) = 0$ (M, discriminant stated) $36 - 52 + 4k = 0 \to$ $k = 4$ (A) Repeated root: $x = 6/2 = 3$; $y = 2(3) + 4 = 10 \to$ contact at $(3, 10)$ (A)

Four marks, three of them earned before any answer appears, the anatomy of method-mark working.

Common mistakes in this topic

• Halving $b$ incorrectly when completing the square with $a \ne 1$ (factor out first, always)
• Discriminant applied without first collecting the quadratic to "$= 0$"
• Inequality regions chosen by memory instead of sketch, the classic wrong-way-round answer
• Tangent questions solved for intersection but never set to equal roots
• Range stated from the original form instead of the completed square

Quadratics reward drilling more than almost any topic, and weaknesses here surface everywhere else in the paper, which is why it’s week 3 of the 8-week revision plan. The must-know forms are on the formula list.

If the discriminant questions keep mutating faster than you can pattern-match them, that’s precisely what a weekly 1-to-1 fixes, free 1-hour trial with Teacher Rig, booked on WhatsApp.