Understanding the Discriminant

In algebra, the discriminant gives us key information about the number of solutions to a quadratic equation.

It’s the expression found under the square root: b² - 4ac. By evaluating the discriminant, we can determine whether a quadratic equation has two real solutions, one real solution, or no real solutions at all. This simple calculation gives powerful insight before solving the equation—just by looking at the numbers!

What Does The Discriminant Tell Us?

  • Positive Discriminant

    Indicates two real solutions.

  • Negative Discriminant

    Indicates no real solutions (complex or imaginary roots)

  • Zero Discriminant

    Indicates one real (repeated) solution

The Discriminant in Action

What are some examples of how we might use the discriminant?

  • Before Solving a Quadratic Equation: For x² + 4x + 5, the discriminant turns out to be -4. Since it’s negative, there are no real solutions—so don’t bother factoring or completing the square.

  • Number of Solutions in a Word Problem: A ball is thrown upward—will it hit the ground again? Use the discriminant to find out if the height equation has real roots.

  • Quick Graph Checks: If a quadratic equation has a negative discriminant, the parabola doesn’t touch the x-axis. That tells you there are no x-intercepts.

Solving the Discriminant (Example #1)

Two Real Solutions

Equation:
x² - 5x + 6 = 0

Step 1: Identify a, b, and c
a = 1, b = -5, c = 6

Step 2: Plug into the discriminant formula
D = b² - 4ac
D = (-5)² - 4(1)(6)
D = 25 - 24
D = 1

Step 3: Interpret the result
Since D > 0, there are two real and distinct solutions.

Solving the Discriminant (Example #2)

One Real Solution

Equation:
x² + 6x + 9 = 0

Step 1: Identify a, b, and c
a = 1, b = 6, c = 9

Step 2: Plug into the discriminant formula
D = b² - 4ac
D = (6)² - 4(1)(9)
D = 36 - 36
D = 0

Step 3: Interpret the result
Since D = 0, there is exactly one real solution (a repeated root).

Solving the Discriminant (Example #3)

No Real Solutions

Equation:
2x² + 4x + 5 = 0

Step 1: Identify a, b, and c
a = 2, b = 4, c = 5

Step 2: Plug into the discriminant formula
D = b² - 4ac
D = (4)² - 4(2)(5)
D = 16 - 40
D = -24

Step 3: Interpret the result
Since D < 0, there are no real solutions—only complex roots.

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