The discriminant and the nature of the roots

What the discriminant is

• The discriminant is the expression under the square root in the quadratic formula:

$$\Delta = b^2 - 4ac$$

• It tells you how many real roots a quadratic has, and what kind — without solving anything.
• This is why questions ask about "the nature of the roots": they want the classification, not the roots themselves.

The three cases

• Note the vocabulary. "Two distinct real roots" means $\Delta > 0$; "equal roots", "a repeated root" or "a double root" all mean $\Delta = 0$; "no real roots" means $\Delta < 0$.
• At Level 2, $\Delta < 0$ means there are no real solutions and you stop there. Complex roots are Level 3.

Why it works

• The quadratic formula is $x = \dfrac{-b \pm \sqrt{\Delta}}{2a}$.
  • If $\Delta > 0$, the square root is a real number and the $\pm$ produces two different answers.
  • If $\Delta = 0$, the root is 0 and $\pm 0$ gives the same answer twice.
  • If $\Delta < 0$, the square root of a negative number is not real, so there are no real solutions.

The graphical meaning

• The roots of $ax^2 + bx + c = 0$ are the $x$-intercepts of the parabola $y = ax^2 + bx + c$.

• $\Delta$ therefore counts the crossings, and this is often the fastest way to interpret a question about a curve and a line.

• A tangent means a repeated root. If a line just touches a parabola, substituting one into the other gives a quadratic with $\Delta = 0$. That connection turns geometry questions into discriminant questions.

Questions with an unknown constant

• The characteristic Merit/Excellence form: "find the values of $k$ for which $x^2 + kx + 9 = 0$ has equal roots".

  • Write the discriminant in terms of $k$.
  • Apply the condition ($> 0$, $= 0$ or $< 0$).
  • Solve the resulting equation or inequality.

• Worked through:

  • $$\Delta = k^2 - 36$$
  • Equal roots means $\Delta = 0$: $k^2 = 36$, so $k = \pm 6$
  • Both values. A squared unknown gives two answers.

• If the condition is an inequality, the answer is a range.

  • "Two distinct real roots" gives $k^2 - 36 > 0$, so $k < -6$ or $k > 6$.
  • "No real roots" gives $k^2 - 36 < 0$, so $-6 < k < 6$.

• Note how the two answers differ in shape: $k^2 > 36$ splits into two separate intervals, while $k^2 < 36$ is a single interval between the roots. Sketching $y = k^2 - 36$ makes the difference obvious.

Watch for a hidden rearrangement

• The discriminant applies only in the form $ax^2 + bx + c = 0$.
• If the question gives $x^2 + kx = 4 - 2x$, rearrange first: $x^2 + (k+2)x - 4 = 0$, and then $b = k + 2$, not $k$.

The 2025 report

• "Used the discriminant to solve problems" is listed as a Merit behaviour, and "linked the discriminant to a problem and substituted to find both values" as Excellence.
• Meanwhile Not Achieved candidates "displayed little understanding of the process for completing the square, theories relating to the discriminant, or quadratic formulae".