Definition: The discriminant of ax^2 + bx + c = 0 is b^2 – 4ac; its sign tells whether the equation has two real solutions, one repeated solution, or none.
What does the discriminant do?
It is the expression under the square root in the quadratic formula. You can compute it alone, before solving, to learn how many real solutions to expect.
How do you read it?
- Positive: two different real solutions, and the parabola crosses the x-axis twice.
- Zero: one repeated real solution, and the vertex touches the x-axis.
- Negative: no real solutions, and the parabola floats above or below the axis.
Example. For 2x2 – 7x + 9 = 0, the discriminant is (-7)2 – 4(2)(9) = 49 – 72 = -23, so there are no real solutions. For 4x2 – 20x + 25 = 0, it is 400 – 400 = 0, giving one solution, x = 5/2. For x2 – 3x – 10 = 0, it is 9 + 40 = 49, so there are two solutions.
What mistake is typical?
Squaring b wrongly when b is negative: (-7)2 is 49, not -49. Another is subtracting 4ac when c is negative and forgetting that the term then adds.
Example: x^2 + 6x + 5 = 0 has discriminant 36 – 20 = 16, so two real solutions.