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Algebra 2 Quadratic functions (advanced)

Discriminant and nature of roots

20 practice questions 0 video lessons Theory + worked examples

The Discriminant and Nature of Roots

Common Core Algebra 2 • Standard A-REI.4b • Quadratic Functions

The Discriminant and Nature of Roots is a topic in Quadratic Functions in the Common Core State Standards. It is aligned to Standard A-REI.4b, which requires students to recognize when the quadratic formula gives complex solutions and write them as a plus or minus b i.

The discriminant \(b^2-4ac\) tells the nature of a quadratic's roots: two real, one repeated, or two complex.

Common Core Algebra 2 › Quadratic Functions › The Discriminant and Nature of Roots  —  Standard A-REI.4b

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Theory

The discriminant of \(ax^2+bx+c=0\) is the quantity under the square root in the quadratic formula:

\[D=b^2-4ac.\]
  • \(D>0\): two distinct real roots.
  • \(D=0\): one repeated real root.
  • \(D<0\): two complex conjugate roots.
The discriminant alone reveals the number and type of roots — no need to solve fully.
The discriminant and roots The discriminant determines whether a parabola meets the x-axis twice, once, or not at all. D > 0 D = 0 D < 0 2 roots 1 root 0 real
Two, one, or no \(x\)-intercepts as \(D\) changes sign.
Discriminant D = b² - 4ac Discriminant D = b² - 4ac Discriminant D = b² - 4ac D > 0: two real roots D = 0: one repeated root D < 0: two complex roots
What the discriminant tells you.

The discriminant:

\[D=b^2-4ac\]
the discriminant is b squared minus 4 a c
A perfect-square \(D\) means the roots are rational.

How to use the discriminant

  1. Write the equation as \(ax^2+bx+c=0\).
  2. Compute \(D=b^2-4ac\).
  3. Compare \(D\) with \(0\) to classify the roots.
  4. Set \(D=0\) to find values giving a repeated root.
Example 1 — Two real roots
Describe the roots of \(x^2-5x+6=0\).
Solution

Compute \(D=b^2-4ac\).

\(D\)\(=\)\((-5)^2-4(1)(6)\)
\(=\)\(25-24=1>0\)

\(D>0\): two distinct real roots.

the discriminant is 1, so two real roots
Example 2 — Repeated root
Describe the roots of \(x^2-4x+4=0\).
Solution

Compute the discriminant.

\(D\)\(=\)\(16-16=0\)

\(D=0\): one repeated real root.

the discriminant is 0, so one repeated root
Example 3 — Complex roots
Describe the roots of \(x^2+x+1=0\).
Solution

Compute the discriminant.

\(D\)\(=\)\(1-4=-3<0\)

\(D<0\): two complex conjugate roots.

the discriminant is negative 3, so two complex roots
Example 4 — Find k for a repeated root
For \(x^2+kx+9=0\) to have one repeated root, find \(k\).
Solution

Set \(D=0\).

\(k^2-4(1)(9)\)\(=\)\(0\)
\(k^2\)\(=\)\(36\)
\(k\)\(=\)\(\pm6\)
k equals plus or minus 6

Common pitfalls

Use the signs of \(a,b,c\) carefully — \((-5)^2=25\), not \(-25\).
\(D<0\) means complex, not “no solution.”
Set the equation to \(=0\) first before reading \(a,b,c\).

Frequently asked questions

What is the discriminant?

\(D=b^2-4ac\), the quantity under the root in the quadratic formula.

What does a positive discriminant mean?

Two distinct real roots.

What does a zero discriminant mean?

One repeated real root — the parabola touches the \(x\)-axis.

What does a negative discriminant mean?

Two complex conjugate roots and no \(x\)-intercepts.

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