The Discriminant and the Type of Solutions

How the sign of b squared minus 4ac reveals whether a quadratic has one repeated root, two complex roots or two real roots, and what each case looks like on the graph.

--

One small calculation tells you what kind of answer a quadratic will give — before you solve it at all. That number is the part hiding under the square root sign, and it is called the discriminant.

Theorem Where it comes from
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
The expression under the root is the discriminant:
\Delta = b^2 - 4ac

Its sign decides everything: whether the roots are real or complex, and whether they are distinct or repeated.

Concept Three pictures
Δ > 0 Δ = 0 Δ < 0 crosses twice touches once never meets
The sign of \Delta matches what the curve does at the x-axis: two crossings, a single touch, or no contact at all.
Example Case 1 — the discriminant is zero

Take x^2 - 4x + 4 = 0 , so a = 1, b = -4, c = 4 .

\Delta = (-4)^2 - 4(1)(4) = 16 - 16 = 0
The root term vanishes, leaving one value.
x = \frac{4}{2} = 2
⟹ one repeated solution, x = 2

Strictly there are still two solutions — they simply happen to be equal. The parabola's vertex sits exactly on the x-axis.

Example Case 2 — the discriminant is negative

Take x^2 + 2x + 5 = 0 , so a = 1, b = 2, c = 5 .

\Delta = (2)^2 - 4(1)(5) = 4 - 20 = -16
A negative under the root brings in i :
\sqrt{-16} = 4i
x = \frac{-2 \pm 4i}{2} = -1 \pm 2i
⟹ two complex conjugate solutions

The curve never reaches the x-axis, which is exactly why no real solution exists.

Example Case 3 — the discriminant is positive

Here there are always two different real solutions, but their type depends on whether \Delta is a perfect square.

Perfect square — x^2 - 5x + 6 = 0 :
\Delta = 25 - 24 = 1 , and \sqrt{1} = 1
x = \frac{5 \pm 1}{2}
⟹ x = 3 or x = 2, both rational
Not a perfect square — x^2 - 3x + 1 = 0 :
\Delta = 9 - 4 = 5 , and \sqrt{5} is irrational
⟹ x = (3 ± √5) / 2, both irrational
Note The four outcomes at a glance
\Delta = 0 — one repeated rational solution; the curve touches the axis.
\Delta < 0 — two complex conjugates; the curve misses the axis.
\Delta > 0 and a perfect square — two rational solutions.
\Delta > 0 , not a perfect square — two irrational solutions containing a root.
Summary
  1. The discriminant is Δ = b² − 4ac, the part under the root.
  2. Δ = 0 gives one repeated solution and a curve touching the x-axis.
  3. Δ < 0 gives two complex conjugates and no crossing.
  4. Δ > 0 gives two distinct real solutions and two crossings.
  5. A perfect-square Δ makes those solutions rational; otherwise they are irrational.
231. Arithmetic Sequences and Series
← Back to Semester