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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

Texas Algebra II (TEKS) • Standard 2A.4(H) • Quadratic Functions

The Discriminant and Nature of Roots is a topic in Quadratic Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.4(H), which requires students to solve quadratic equations, including those with complex solutions.

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

Texas Algebra II (TEKS) › Quadratic Functions › The Discriminant and Nature of Roots  —  Standard 2A.4(H)

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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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