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.

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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
The expression under the root is the discriminant:

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 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 , so .

The root term vanishes, leaving one value.
⟹ 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 , so .

A negative under the root brings in :
⟹ 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 is a perfect square.

Perfect square — :
, and
⟹ x = 3 or x = 2, both rational
Not a perfect square — :
, and is irrational
⟹ x = (3 ± √5) / 2, both irrational
Note The four outcomes at a glance
— one repeated rational solution; the curve touches the axis.
\Delta &lt; 0 — two complex conjugates; the curve misses the axis.
\Delta &gt; 0 and a perfect square — two rational solutions.
\Delta &gt; 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.
283. Relations and Functions
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