Kid-friendly explanation
When you solve a quadratic equation, the discriminant is a special number you calculate from a, b, and c. It tells you what kind of answers to expect before you do all the work. If the discriminant is positive, you get 2 different real answers. If it is 0, you get 1 real answer, and if it is negative, you get 2 not-real (complex) answers.
Worked examples
Example 1 (simpler): Decide how many real solutions a quadratic has
Consider the quadratic equation 2x² + 3x − 5 = 0.
- Identify a, b, and c. Here a = 2, b = 3, and c = −5.
- The discriminant is b² − 4ac.
- Compute b²: 3² = 9.
- Compute 4ac: 4 × 2 × (−5) = 8 × (−5) = −40.
- Now find b² − 4ac: 9 − (−40) = 9 + 40 = 49.
- The discriminant is 49, which is positive.
Because the discriminant is positive, this quadratic equation has 2 different real solutions.
Example 2 (more advanced): Use the discriminant to describe and approximate solutions
Consider the quadratic equation 3x² − 6x + 5 = 0.
- Identify a, b, and c. Here a = 3, b = −6, and c = 5.
- The discriminant is b² − 4ac.
- Compute b²: (−6)² = 36.
- Compute 4ac: 4 × 3 × 5 = 12 × 5 = 60.
- Now find b² − 4ac: 36 − 60 = −24.
- The discriminant is −24, which is negative.
Because the discriminant is negative, this quadratic equation has 2 not-real (complex) solutions and no real solutions. If you used the quadratic formula, you would see a square root of −24, which involves the imaginary unit i.
Going deeper with the discriminant
The discriminant is closely tied to the graph of a quadratic function, which is a U-shaped curve called a parabola. When the discriminant is positive, the parabola crosses the x-axis at 2 points. When it is 0, the parabola just touches the x-axis at 1 point, called a repeated root. When it is negative, the parabola does not touch the x-axis at all, which matches the idea that there are no real solutions.
In algebra courses, students often use the discriminant to quickly decide whether it is worth trying to factor a quadratic, to check whether a quadratic has real solutions, and to describe how many solutions an equation has. Later, in more advanced math, the same idea appears in other types of equations and helps classify different kinds of functions and curves.
Common mistakes with the discriminant
- Mixing up the formula. A common error is to write the discriminant as 4ac − b² instead of b² − 4ac. This changes the sign and can turn a positive discriminant into a negative one, so always start with b² and then subtract 4ac.
- Forgetting to square the negative sign on b. For b = −5, some students compute b² as 5² = 25 instead of (−5)² = 25, which is correct by luck here, but with other steps it can lead to sign errors. Always treat b as a whole number, including its sign, when squaring.
- Misreading what the discriminant tells you. Some students think a discriminant of 0 means "no solution". In fact, a discriminant of 0 means there is exactly 1 real solution, because the 2 solutions from the quadratic formula are the same number.
How the discriminant shows up in everyday life
The discriminant itself is a math tool, but the quadratic equations it comes from model many real situations. For example, when you track the height of a ball thrown into the air, its path can be described by a quadratic equation. The discriminant of that equation can tell you whether the ball ever reaches a certain height or when it hits the ground, by showing whether there are real times that make the equation true.