Discriminant Formula

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

The Formula

ฮ”=b2โˆ’4ac\Delta = b^2 - 4ac
ฮ”>0\Delta > 0: two distinct real solutions.
ฮ”=0\Delta = 0: exactly one real solution (double root).
ฮ”<0\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 x2โˆ’5x+6=0x^2 - 5x + 6 = 0: ฮ”=(โˆ’5)2โˆ’4(1)(6)=25โˆ’24=1>0\Delta = (-5)^2 - 4(1)(6) = 25 - 24 = 1 > 0 Two distinct real solutions.

Notation

ฮ”\Delta (Greek letter delta) denotes the discriminant. It is the expression under the x\sqrt{\phantom{x}} in the quadratic formula: ฮ”=b2โˆ’4ac\sqrt{\Delta} = \sqrt{b^2 - 4ac}.

What This Formula Means

The discriminant of a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 is the expression ฮ”=b2โˆ’4ac\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 ax2+bx+c=0ax^2 + bx + c = 0, define ฮ”=b2โˆ’4ac\Delta = b^2 - 4ac. Then: ฮ”>0โ‡’\Delta > 0 \Rightarrow two distinct real roots; ฮ”=0โ‡’\Delta = 0 \Rightarrow one repeated real root (r=โˆ’b2ar = -\frac{b}{2a}); ฮ”<0โ‡’\Delta < 0 \Rightarrow two conjugate complex roots r=โˆ’bยฑiโˆฃฮ”โˆฃ2ar = \frac{-b \pm i\sqrt{|\Delta|}}{2a}.

Worked Examples

Example 1

easy
Find the discriminant of x2โˆ’6x+9=0x^2 - 6x + 9 = 0 and determine the number of solutions.

Answer

ฮ”=0\Delta = 0; one repeated solution (x=3x = 3).

First step

1
Identify a=1,b=โˆ’6,c=9a = 1, b = -6, c = 9.

Full solution

  1. 2
    Discriminant: ฮ”=b2โˆ’4ac=36โˆ’36=0\Delta = b^2 - 4ac = 36 - 36 = 0.
  2. 3
    Since ฮ”=0\Delta = 0, there is exactly one real solution (a repeated root).
The discriminant ฮ”=b2โˆ’4ac\Delta = b^2 - 4ac tells us the nature of the roots: ฮ”>0\Delta > 0 means two distinct real roots, ฮ”=0\Delta = 0 means one repeated root, ฮ”<0\Delta < 0 means no real roots.

Example 2

medium
For what values of kk does x2+kx+9=0x^2 + kx + 9 = 0 have two distinct real solutions?

Example 3

medium
Use the discriminant to decide whether 3x2+7xโˆ’6=03x^2 + 7x - 6 = 0 has two real solutions.

Common Mistakes

  • Computing b2โˆ’4acb^2-4ac with the wrong sign on bb - b2b^2 is always nonnegative, so square the value of bb carefully (e.g. (โˆ’5)2=25(-5)^2=25).
  • Dropping the โˆ’4ac-4ac - the discriminant is the whole b2โˆ’4acb^2-4ac, not just b2b^2.
  • Saying 'no solutions' when ฮ”<0\Delta<0 - there are no REAL solutions, but two complex ones.

Why This Formula Matters

It answers 'how many real solutions?' in one cheap computation, which is exactly what graphing (does the parabola cross the x-axis?) and existence questions need. It saves you from solving when you only need to know whether a solution exists. Recognizing it by "Do I only need to know how many/what kind of roots a quadratic has, not their values?" โ€” rather than by familiar numbers โ€” is what lets a student tell it apart from quadratic formula and zeros of a quadratic and vertex/min-max in a mixed problem set.

Frequently Asked Questions

What is the Discriminant formula?

The discriminant of a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 is the expression ฮ”=b2โˆ’4ac\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?

ฮ”\Delta (Greek letter delta) denotes the discriminant. It is the expression under the x\sqrt{\phantom{x}} in the quadratic formula: ฮ”=b2โˆ’4ac\sqrt{\Delta} = \sqrt{b^2 - 4ac}.

Why is the Discriminant formula important in Math?

It answers 'how many real solutions?' in one cheap computation, which is exactly what graphing (does the parabola cross the x-axis?) and existence questions need. It saves you from solving when you only need to know whether a solution exists. Recognizing it by "Do I only need to know how many/what kind of roots a quadratic has, not their values?" โ€” rather than by familiar numbers โ€” is what lets a student tell it apart from quadratic formula and zeros of a quadratic and vertex/min-max in a mixed problem set.

What do students get wrong about Discriminant?

The procedure for discriminant is the easy part; the trap is computing b2โˆ’4acb^2-4ac with the wrong sign on bb. Asking "Do I only need to know how many/what kind of roots a quadratic has, not their values?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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 โ†’
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