Discriminant & Nature of Roots

For $ax^2 + bx + c = 0$, the discriminant $b^2 - 4ac$ decides how many real roots exist:

• $b^2 - 4ac > 0$ $\to$ two distinct real roots
• $b^2 - 4ac = 0$ $\to$ two equal (repeated) roots, the graph touches the axis
• $b^2 - 4ac < 0$ $\to$ no real roots, the graph never reaches the axis

This three-line table powers more 0606 marks than almost any other single fact, because the exam wraps it in algebra rather than asking it straight.

The find-k patterns

Equal roots: ”$kx^2 + 12x + 9 = 0$ has equal roots; find $k$.” $b^2 - 4ac = 0$ (write this line, it’s the M mark): $144 - 36k = 0 \to$ $k = 4$.

Real roots / no real roots: the same setup with an inequality, ending in a quadratic or linear inequality in k. “Real roots” means $b^2 - 4ac$ $\ge$ $0$, equal roots are still real; the $\ge$/$>$ distinction is a deliberate trap. “Two distinct real roots” is the phrase that means strictly $>$.

Always positive: “Show that $x^2 + 2kx + k^2 + 3 > 0$ for all $x$.” Two accepted arguments: discriminant $< 0$ ($4k^2 - 4k^2 - 12 = -12 < 0$) plus $a > 0$, stated together; or complete the square to $(x + k)^2 + 3 \ge 3 > 0$. Either way, the conclusion sentence in words carries the final mark.

Lines and curves: the geometric costume

Substitute the line into the curve, collect to "$= 0$", and the discriminant reads the geometry: two intersections, tangent ($= 0$), or miss ($< 0$). The full routine, including the bracket discipline of the substitution line, is in line–curve intersection, and the same machinery extends to circles.

Working that banks the marks

1. Identify $a$, $b$, $c$ after collecting everything to one side, a misidentified $b$ from an uncollected equation is the most common wrong answer.
2. Write the discriminant condition as its own line: “for equal roots, $b^2 - 4ac = 0$”.
3. Substitute, solve, and conclude in words when the question asked a question (“since $b^2 - 4ac = -8 < 0$, the line does not meet the curve”).

Common mistakes

• $a$, $b$, $c$ read off before rearranging to $= 0$
• $\ge$ vs $>$ blurred (“real” vs “distinct”)
• $b^2 - 4ac$ computed but the conclusion never stated
• Sign slips squaring negative $b$ ($b^2$ is always positive)

Full topic context: Quadratic Functions notes.