Discriminant

Algebra
definition

Also known as: delta, b squared minus 4ac

Grade 9-12

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

This concept is covered in depth in our quadratic formula and discriminant guide, with worked examples, practice problems, and common mistakes.

Definition

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.

💡 Intuition

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.

🎯 Core Idea

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

Example

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

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).

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}.

🌟 Why It 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.

💭 Hint When Stuck

Write out b^2 and 4ac separately before subtracting to avoid sign errors.

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}.

🚧 Common Stuck Point

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

⚠️ 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

Frequently Asked Questions

What is Discriminant in Math?

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.

Why is Discriminant important?

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 usually 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 Discriminant?

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

How Discriminant Connects to Other Ideas

To understand discriminant, you should first be comfortable with quadratic formula and quadratic standard form. Once you have a solid grasp of discriminant, you can move on to zeros of quadratic and complex numbers.

Want the Full Guide?

This concept is explained step by step in our complete guide:

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