When we look at quadratic equations, one important part to think about is the discriminant. We write it as ( D = b^2 - 4ac ). This number can tell us a lot about the solutions of the equation, like how many roots it has and what they are like. Let's go through this based on what the discriminant value is.

1. Positive Discriminant (( D > 0 ))

If the discriminant is positive, it means that the quadratic equation has two different real roots. This means if you were to graph it, the curve would cross the x-axis at two points.

For example, let's look at this equation:

( x^2 - 5x + 6 = 0 )

In this case, we have ( a = 1 ), ( b = -5 ), and ( c = 6 ).

Now, let’s calculate the discriminant:

[ D = (-5)^2 - 4(1)(6) = 25 - 24 = 1 ]

Since ( D > 0 ), we know there are two different roots.

2. Zero Discriminant (( D = 0 ))

When the discriminant is zero, there is exactly one real root, which we call a repeated or double root. This means the graph just touches the x-axis at one point but doesn’t go across it.

For example, consider this equation:

( x^2 - 4x + 4 = 0 )

Here, ( a = 1 ), ( b = -4 ), and ( c = 4 ).

Calculating the discriminant gives us:

[ D = (-4)^2 - 4(1)(4) = 16 - 16 = 0 ]

Since ( D = 0 ), we find that there’s one real root, which is ( x = 2 ).

3. Negative Discriminant (( D < 0 ))

A negative discriminant means that the quadratic has no real roots. Instead, it has two complex roots. The graph does not touch or cross the x-axis at all.

For example, take this equation:

( x^2 + 4x + 8 = 0 )

Here, ( a = 1 ), ( b = 4 ), and ( c = 8 ).

Calculating the discriminant:

[ D = (4)^2 - 4(1)(8) = 16 - 32 = -16 ]

Since ( D < 0 ), it tells us there are no real solutions, and the roots are complex.

Summary

To wrap it up:

• ( D > 0 ): Two different real roots.
• ( D = 0 ): One real double root.
• ( D < 0 ): Two complex roots.

Understanding the discriminant helps us learn more about quadratic equations and what kinds of solutions they have!