Discriminant Formula

The Formula

\Delta = b^2 - 4ac
\Delta > 0: two distinct real solutions.
\Delta = 0: exactly one real solution (double root).
\Delta < 0: no real solutions (two complex solutions).

When to use: 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.

Quick Example

For x^2 - 5x + 6 = 0: \Delta = (-5)^2 - 4(1)(6) = 25 - 24 = 1 > 0 Two distinct real solutions.

Notation

\Delta (Greek letter delta) denotes the discriminant. It is the expression under the \sqrt{\phantom{x}} in the quadratic formula: \sqrt{\Delta} = \sqrt{b^2 - 4ac}.

What This Formula Means

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.

Formal View

For ax^2 + bx + c = 0, define \Delta = b^2 - 4ac. Then: \Delta > 0 \Rightarrow two distinct real roots; \Delta = 0 \Rightarrow one repeated real root (r = -\frac{b}{2a}); \Delta < 0 \Rightarrow two conjugate complex roots r = \frac{-b \pm i\sqrt{|\Delta|}}{2a}.

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?

Common Mistakes

  • Squaring b incorrectly when b is negative (the square is always positive)
  • Forgetting the factor of 4 in 4ac
  • Misreading the coefficients—make sure the equation is in standard form ax^2 + bx + c = 0 first

Why This Formula Matters

Saves time by revealing whether factoring will work (perfect square discriminant), whether the parabola touches the x-axis, and whether real solutions exist at all.

Frequently Asked Questions

What is the Discriminant formula?

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.

How do you use the Discriminant formula?

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.

What do the symbols mean in the Discriminant formula?

\Delta (Greek letter delta) denotes the discriminant. It is the expression under the \sqrt{\phantom{x}} in the quadratic formula: \sqrt{\Delta} = \sqrt{b^2 - 4ac}.

Why is the Discriminant formula important in Math?

Saves time by revealing whether factoring will work (perfect square discriminant), whether the parabola touches the x-axis, and whether real solutions exist at all.

What do students get wrong about Discriminant?

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

What should I learn before the Discriminant formula?

Before studying the Discriminant formula, you should understand: quadratic formula, quadratic standard form.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Quadratic Equations: Factoring, Completing the Square, and the Quadratic Formula →
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