To find the real and complex roots of a quadratic equation, we can use a method called the discriminant.
The quadratic formula looks like this:
[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]
Here, ( a ), ( b ), and ( c ) are numbers that we use in the equation.
Step 1: Calculate the Discriminant
The discriminant is represented by the letter ( D ). You calculate it like this:
[ D = b^2 - 4ac ]
Step 2: Determine the Nature of the Roots
Now, let's see what the value of ( D ) tells us about the roots:
Example 1:
For the equation ( x^2 - 4x + 4 = 0 ):
Here, ( a = 1 ), ( b = -4 ), and ( c = 4 ).
Example 2:
For the equation ( x^2 + 1 = 0 ):
Calculate ( D ):
[ D = 0^2 - 4(1)(1) = -4 ]
This means there are two complex roots, which are ( x = i ) and ( x = -i ).
And that’s how you use the discriminant to identify the roots of a quadratic equation!
To find the real and complex roots of a quadratic equation, we can use a method called the discriminant.
The quadratic formula looks like this:
[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]
Here, ( a ), ( b ), and ( c ) are numbers that we use in the equation.
Step 1: Calculate the Discriminant
The discriminant is represented by the letter ( D ). You calculate it like this:
[ D = b^2 - 4ac ]
Step 2: Determine the Nature of the Roots
Now, let's see what the value of ( D ) tells us about the roots:
Example 1:
For the equation ( x^2 - 4x + 4 = 0 ):
Here, ( a = 1 ), ( b = -4 ), and ( c = 4 ).
Example 2:
For the equation ( x^2 + 1 = 0 ):
Calculate ( D ):
[ D = 0^2 - 4(1)(1) = -4 ]
This means there are two complex roots, which are ( x = i ) and ( x = -i ).
And that’s how you use the discriminant to identify the roots of a quadratic equation!