1. Let’s see why the answers (1) and (2) are the same
Roots of Equations
Roots is just another word for solutions !
e.g. Find the roots of the equation
Solution: There are no factors, so we can either complete the square or use the quadratic formula.
Completing the square:
Using the formula:
Let’s see why the answers (1) and (2) are the same

2. The answers from the quadratic formula can be simplified:
We have
Numbers such as are called surds
However,
4 is a perfect square so can be square-rooted, so
We have simplified the surd
So,
2 is a common factor of the numerator, so

3. Exercise
Simplify the following surds:

4. The Discriminant of a Quadratic Function
The formula for solving a quadratic equation is
The part is called the discriminant
Because we square root the discriminant, we get different types of roots depending on its sign.

5. The Discriminant of a Quadratic Function
we consider the graph of the function
To investigate the roots of the equation
The roots of the equation are at the points where y = 0
( x = 1 and x = 4)
The discriminant
The roots are real and distinct.
( different )

6. The Discriminant of a Quadratic Function
For the equation
the discriminant
The roots are real and equal
( x = 2)

7. The Discriminant of a Quadratic Function
For the equation
the discriminant
There are no real roots as the function is never equal to zero
If we try to solve , we get
The square of any real number is positive so there are no real solutions to

8. SUMMARY
The formula for solving the quadratic equation is
The part is called the discriminant
The roots are real and distinct
( different )
The roots are real and equal
The roots are not real
If we try to solve an equation with no real roots, we will be faced with the square root of a negative number!

9. Exercise
1 (a) Use the discriminant to determine the nature of the roots of the following quadratic equations:
(i)
(ii)
(b) Check your answers by completing the square to find the vertex of the function and sketching.
Solution: (a) (i)
The roots are real and equal.
(ii)
The roots are real and distinct.

10. (b) Check your answers by completing the square to find the vertex of the function and sketching.
(b) (i)
 Vertex is ( -1,0 )
Roots of equation
(real and equal)
(ii)
 Vertex is ( 1,-2 )
Roots of equation
(real and distinct)

11. 2. Determine the nature of the roots of the following quadratic equations ( real and distinct or real and equal or not real ) by using the discriminant. DON’T solve the equations.
(a)
Roots are real and equal
(b)
There are no real roots
(c)
Roots are real and distinct