To find out how many roots a quadratic equation has, we look at something called the discriminant. A quadratic equation can be written like this:

$ax^2 + bx + c = 0$

Here, ( a ), ( b ), and ( c ) are real numbers, and ( a ) cannot be zero.

The discriminant, which we call ( D ), is found using this formula:

$D = b^2 - 4ac$

The value of ( D ) tells us important details about the roots of the quadratic equation.

Types of Roots Based on the Discriminant

1. Two Different Real Roots:

   • When: If ( D > 0 )
   • What It Means: This means that the quadratic curve touches the x-axis at two places, which gives two different real roots.
   • Example: Take the equation ( x^2 - 5x + 6 = 0 ).
     • We calculate ( D = (-5)^2 - 4(1)(6) = 25 - 24 = 1 ).
     • Since ( D > 0 ), there are two different real roots: ( x = \frac{5 \pm 1}{2} = 3 ) and ( 2 ).

2. One Real Root (Repeated):

   • When: If ( D = 0 )
   • What It Means: Here, the quadratic touches the x-axis at just one point. This point is a repeated root (or double root).
   • Example: Look at ( x^2 - 4x + 4 = 0 ).
     • We have ( D = (-4)^2 - 4(1)(4) = 16 - 16 = 0 ).
     • This tells us there’s one real root, specifically ( x = \frac{4}{2} = 2 ), which counts twice.

3. No Real Roots:

   • When: If ( D < 0 )
   • What It Means: A negative value for ( D ) means that the quadratic does not touch or cross the x-axis. This results in two complex (imaginary) roots.
   • Example: Consider ( x^2 + x + 1 = 0 ).
     • We find ( D = 1^2 - 4(1)(1) = 1 - 4 = -3 ).
     • So, there are no real roots, and the solutions are complex numbers, calculated as ( x = \frac{-1 \pm \sqrt{-3}}{2} = -\frac{1}{2} \pm \frac{\sqrt{3}}{2}i ).

In short, the discriminant helps us quickly understand the types of roots a quadratic equation has:

• ( D > 0 ): Two different real roots
• ( D = 0 ): One real root (double root)
• ( D < 0 ): No real roots (two complex roots)