The document discusses the discriminant and how it is used to determine the number of solutions to a quadratic equation. It defines the discriminant as b^2 - 4ac and explains that: - If the discriminant is positive, there are 2 solutions - If the discriminant is 0, there is 1 solution - If the discriminant is negative, there are no real solutions. Several examples are provided to demonstrate calculating the discriminant and using it to determine the nature of the roots. The key aspects are to identify the coefficients a, b, and c and then evaluate and compare b^2 - 4ac to determine how many solutions exist.

1. Block 2
The Discriminant

2. What is to be learned?
• What the discriminant is
• How we use the discriminant to find out
how many solutions there are (if any)

3. Quadratic Formula
x = -b +
- √b2 – 4ac
2a
How many solutions?
1. x2 + 6x + 9 = 0
2. x2 + 8x + 9 = 0
3. x2 + 4x + 9 = 0
1
2
0

4. The Big Nasty Formula
2a
x = -b +
- √b2 – 4ac
How many solutions?
1. x2 + 6x + 9 = 0
2. x2 + 8x + 9 = 0
3. x2 + 4x + 9 = 0
1
2
0
The Discriminant
tells you how many
solutions you have
zero
positive
negative

5. The Discriminant
b2 – 4ac
If b2 – 4ac is positive
If b2 – 4ac is zero
If b2 – 4ac is negative  0 solutions
 2 solutions
 1 solution

6. The Discriminant
b2 – 4ac
If b2 – 4ac is positive
If b2 – 4ac is zero
If b2 – 4ac is negative  0 solutions
Or
 2 solutions
 1 solution

7. The Discriminant
b2 – 4ac
If b2 – 4ac > 0
If b2 – 4ac = 0
If b2 – 4ac < 0
 2 solutions
 1 equal roots
 Imaginary roots
Or

8. Nature of Roots
x2+ 5x – 11 = 0
c.f. ax2 + bx + c = 0
a = 1, b = 5, c = -11
Using Discriminant
b2– 4ac
= 52 – 4(1)(-11)
= 25 + 44
= 69
2 real roots

9. The Discriminant
The b2 – 4ac part of the BNF
If b2 – 4ac > 0  2 solutions ( 2 real roots)
If b2 – 4ac = 0  1 solution(1 real root)
(equal roots)
If b2 – 4ac < 0  0 solutions (no real roots)

10. Nature of Roots
x2+ 6x + 10 = 0
c.f. ax2 + bx + c = 0
a = 1, b = 6, c = 10
Using Discriminant
b2– 4ac
= 62 – 4(1)(10)
= - 4
no real roots
Must be written
like this

11. Key Question
Find the nature of the roots of this equation
3m(m + 2) + 4m = 7
3m2 + 6m + 4m = 7
3m2 + 10m – 7 = 0
c.f. am2 + bm + c = 0
a = 3, b = 10, c = -7
Using Discriminant
b2– 4ac
= 102 – 4(3)(-7)
= 100 + 84
= 184

12. Using The Discriminant
If equal Roots find value of t.
tx2+ 8x + 4 = 0
c.f. ax2 + bx + c = 0
a = t, b = 8,
For equal roots
64 – 16t = 0
64 = 16t
t = 4
b2 – 4ac = 0
82 – 4t(4) = 0
c = 4
State Rule
Get Values
Sub Values
Solve

13. 64 – 8g = 0
64 = 8g
g = 8
c.f. ax2 + bx + c = 0
a = 2, b = -8, c = g
For equal roots
b2 – 4ac = 0
(-8)2 – 4(2)g = 0
Using The Discriminant
If equal Roots find value of g.
2x2– 8x + g = 0
State Rule
Get Values
Sub Values
Solve

14. Key Question
If equal Roots find value of r.
rx2– 18x + 27 = 0
c.f. ax2 + bx + c = 0
a = r, b = -18,c = 27
For equal roots
(-18)2 – 4r(27) = 0
324 – 108r = 0
324 = 108r
r = 3
b2 – 4ac = 0

15. Equal roots
2x2 + (m+1)x + 8 = 0
c.f. ax2 + bx + c = 0
a = 2, b = m+1, c = 8
For equal roots
m2 + 2m – 63 = 0
(m + 9)(m – 7) = 0
m = -9 or m = 7
b2 – 4ac = 0
(m+1)2 – 4(2)(8) = 0
m2 + 2m + 1 – 64 = 0
(m+1)(m+1)
=m2 +2m+1
Quadratic Equation

16. No real roots?
2x2– 8x + g = 0
c.f. ax2 + bx + c = 0
a = 2, b = -8, c = g
For no real roots b2 – 4ac < 0
(-8)2 – 4(2)g < 0
64 – 8g < 0
Inequation

17. Solving Equations
- Reminder
4x = 12 4x > 12
x = 3 x > 3 (Solution)
4x – 2 = 10
Solving Inequations
4x – 2 >
= 10
Main difference is the sign in the “middle”
One other bigdifference

18. 10 > 6
Add 4 to each side
10 + 4 > 6 + 4
14 > 10
True

19. 10 > 6
Multiply each side by 3
10 X 3 > 6 X 3
30 > 18
True

20. 10 > 6
Divide each side by 2
10 ÷ 2 > 6 ÷ 2
5 > 3
True

21. 10 > 6
Divide each side by -2
10÷(-2) > 6÷(-2)
-5 -3
False!!!!!!!
>
If dividing/multiplying by a negative
you must turn sign round
Sorted!

22. Back to….
64 – 8g < 0
– 8g < -64
g > 8
No real roots if g > 8