Discriminant Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Discriminant.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The discriminant of a quadratic equation ax^2 + bx + c = 0 is the expression \Delta = b^2 - 4ac. It determines the number and nature of the solutions.

The discriminant is the expression under the square root in the quadratic formula. If it is positive, you can take the square root and get two answers. If it is zero, the square root is zero so both answers are the same. If it is negative, you cannot take a real square root, so there are no real solutions.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The discriminant is a quick diagnostic test—before solving, it tells you what kind of answer to expect.

Common stuck point: Remember that b in b^2 - 4ac includes its sign. If b = -5, then b^2 = 25, not -25.

Sense of Study hint: Write out b^2 and 4ac separately before subtracting to avoid sign errors.

Worked Examples

Example 1

easy
Find the discriminant of x^2 - 6x + 9 = 0 and determine the number of solutions.

Solution

  1. 1
    Identify a = 1, b = -6, c = 9.
  2. 2
    Discriminant: \Delta = b^2 - 4ac = 36 - 36 = 0.
  3. 3
    Since \Delta = 0, there is exactly one real solution (a repeated root).

Answer

\Delta = 0; one repeated solution (x = 3).
The discriminant \Delta = b^2 - 4ac tells us the nature of the roots: \Delta > 0 means two distinct real roots, \Delta = 0 means one repeated root, \Delta < 0 means no real roots.

Example 2

medium
For what values of k does x^2 + kx + 9 = 0 have two distinct real solutions?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find the discriminant of 2x^2 + 5x - 3 = 0.

Example 2

medium
Without solving, how many real solutions does 3x^2 + 2x + 5 = 0 have?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

quadratic formulaquadratic standard form