Year 11 Methods (Unit 1 & 2) Quadratics

The Discriminant

Q: What is the discriminant of a quadratic?

For a x squared plus b x plus c equals zero, the discriminant is b squared minus four a c. Its sign tells you how many real solutions the equation has.

Q: How does the discriminant tell the number of solutions?

If the discriminant is greater than zero there are two distinct real solutions, if it equals zero there is one repeated solution, and if it is less than zero there are no real solutions.

Q: How do I tell if the solutions are rational or irrational?

When the discriminant is greater than zero, check whether it is a perfect square. If it is a perfect square the solutions are rational; if not, they are irrational surds.

Q: What does the discriminant say about the graph?

It gives the number of x-intercepts of the parabola. Two intercepts when the discriminant is greater than zero, one touch when it equals zero, and none when it is less than zero.

Q: How do I find a value that makes the roots equal?

Set the discriminant, b squared minus four a c, equal to zero and solve for the unknown coefficient. That is the equal roots, or tangency, condition.

Q: Do I need to solve the equation to use the discriminant?

No. The discriminant is a shortcut that counts and classifies the solutions without finding their values.

20 practice questions 1 video lesson Theory + worked examples

Understand the discriminant for Queensland Year 11 Mathematical Methods (QCAA). The discriminant is the part of the quadratic formula under the square root, and its sign counts the solutions.

You will learn to compute it, decide whether a quadratic has two, one or no real solutions, test whether those solutions are rational or irrational, and find an unknown coefficient for one repeated solution — all linked to the parabola's x-intercepts.

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Theory

In Year 11 Mathematical Methods (QCAA, Unit 1), the discriminant \(\Delta=b^2-4ac\) tells you how many solutions the quadratic \(ax^2+bx+c=0\) has — without solving it. This page shows how to compute \(\Delta\), read off the number and nature of the solutions, find an unknown coefficient for a stated condition, and link \(\Delta\) to the number of x-intercepts of the parabola.

The discriminant of \(ax^2+bx+c=0\) is the expression under the root sign in the quadratic formula, \(\Delta=b^2-4ac\). Its sign decides how many real solutions the equation has.

When \(\Delta>0\) there are two distinct real solutions; when \(\Delta=0\) there is one (a repeated, or equal, solution); when \(\Delta<0\) there are no real solutions.

When \(\Delta>0\) the nature can be refined: if \(\Delta\) is a perfect square the two solutions are rational (the quadratic factorises over the integers); otherwise they are irrational surds.

Sign of \(\Delta\) = number of solutions = number of x-intercepts. Two if \(\Delta>0\), one if \(\Delta=0\), none if \(\Delta<0\).

\(\Delta>0\): the parabola cuts the \(x\)-axis at two points.

\(\Delta=0\): the lower parabola just touches once; \(\Delta<0\): the upper one misses entirely.

For \(ax^2+bx+c=0\), the discriminant is:

\[\Delta=b^2-4ac\]

Δ = b^{2} - 4 a c

The sign of \(\Delta\) gives the number of real solutions:

\[\Delta>0\ \Rightarrow\ 2,\qquad \Delta=0\ \Rightarrow\ 1,\qquad \Delta<0\ \Rightarrow\ 0\]

Δ > 0 \Rightarrow 2, Δ = 0 \Rightarrow 1, Δ < 0 \Rightarrow 0

Equal roots condition: a quadratic has exactly one (repeated) solution — and its graph touches the \(x\)-axis — precisely when \(\Delta=b^2-4ac=0\).

How to use the discriminant

Standardise: write the equation as \(ax^2+bx+c=0\) and read off \(a\), \(b\), \(c\) with their signs.
Compute: evaluate \(\Delta=b^2-4ac\).
Interpret: use the sign of \(\Delta\) for the number of solutions (and a perfect-square test for rational versus irrational), or set the required inequality/equation in \(\Delta\) to find an unknown coefficient.

Example 1 — Number and nature

For \(2x^2+3x-5=0\), find the discriminant and state the number and nature of the solutions.

Solution

Coefficients:

\(a\)	\(=\)	\(2\)
\(b\)	\(=\)	\(3\)
\(c\)	\(=\)	\(-5\)

Compute \(\Delta=b^2-4ac\):

\(\Delta\)	\(=\)	\((3)^2-4(2)(-5)\)
	\(=\)	\(9+40\)
	\(=\)	\(49\)

\(\Delta=49>0\), and \(49=7^2\) is a perfect square.

Two real solutions; since \(\Delta\) is a perfect square, they are rational.

Δ = 49

Example 2 — Irrational solutions

How many solutions does \(x^2+4x+2=0\) have, and are they rational or irrational?

Solution

Coefficients:

\(a\)	\(=\)	\(1\)
\(b\)	\(=\)	\(4\)
\(c\)	\(=\)	\(2\)

Compute the discriminant:

\(\Delta\)	\(=\)	\((4)^2-4(1)(2)\)
	\(=\)	\(16-8\)
	\(=\)	\(8\)

\(\Delta=8>0\), but \(8\) is not a perfect square.

Two real solutions, and they are irrational (surds).

Δ = 8

Example 3 — Equal roots (find k)

Find the values of \(k\) for which \(x^2+kx+9=0\) has exactly one solution.

Solution

One solution means equal roots, so set \(\Delta=0\):

\(a\)	\(=\)	\(1\)
\(b\)	\(=\)	\(k\)
\(c\)	\(=\)	\(9\)

Form \(\Delta=b^2-4ac\) and equate to \(0\):

\(k^2-4(1)(9)\)	\(=\)	\(0\)
\(k^2-36\)	\(=\)	\(0\)
\(k^2\)	\(=\)	\(36\)
\(k\)	\(=\)	\(\pm 6\)

One solution when \(k=6\) or \(k=-6\).

k = \pm 6

Example 4 — No solutions (find c)

For what values of \(c\) does \(2x^2+4x+c=0\) have no real solutions?

Solution

No real solutions means \(\Delta<0\):

\(a\)	\(=\)	\(2\)
\(b\)	\(=\)	\(4\)
\(c\)	\(=\)	\(c\)

Form \(\Delta\) and require it to be negative:

\((4)^2-4(2)(c)\)	\(<\)	\(0\)
\(16-8c\)	\(<\)	\(0\)
\(16\)	\(<\)	\(8c\)
\(2\)	\(<\)	\(c\)

No real solutions when \(c>2\); the parabola then sits entirely above the \(x\)-axis.

c > 2

Common pitfalls

Sign slips in \(b^2-4ac\). Square \(b\) as a whole, so \((-4)^2=16\), and watch the sign of \(c\): with \(c=-5\), \(-4ac=+40\), not \(-40\).

Confusing number with nature. \(\Delta>0\) always gives two solutions; the perfect-square test only decides whether those two are rational or irrational.

Actually solving the equation. The discriminant answers how many and what kind — you do not need the root values, so do not run the whole quadratic formula.

Wrong condition for the task. Equal roots use \(\Delta=0\), two solutions use \(\Delta>0\), and none use \(\Delta<0\) — match the inequality to the words.

Frequently asked questions

What is the discriminant of a quadratic?

For \(ax^2+bx+c=0\) it is \(\Delta=b^2-4ac\); its sign tells you how many real solutions the equation has.

How does the discriminant tell the number of solutions?

\(\Delta>0\) gives two distinct real solutions, \(\Delta=0\) gives one repeated solution, and \(\Delta<0\) gives no real solutions.

How do I tell if the solutions are rational or irrational?

When \(\Delta>0\), check whether \(\Delta\) is a perfect square: if it is, the solutions are rational; if not, they are irrational surds.

What does the discriminant say about the graph?

It gives the number of \(x\)-intercepts of \(y=ax^2+bx+c\): two if \(\Delta>0\), one (a touch) if \(\Delta=0\), none if \(\Delta<0\).

How do I find a value that makes the roots equal?

Set \(\Delta=b^2-4ac=0\) and solve for the unknown coefficient; that is the equal-roots (tangency) condition.

Do I need to solve the equation to use the discriminant?

No — the discriminant is a shortcut that counts and classifies the solutions without finding them.

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The Discriminant

📖 Theory

How to use the discriminant

Common pitfalls

Frequently asked questions

Theory