The document discusses the discriminant and how it is used to determine the number of solutions to a quadratic equation. The discriminant, b^2 - 4ac, is calculated and compared to 0 to find the number of solutions: if greater than 0 there are 2 solutions, if equal to 0 there is 1 solution, if less than 0 there are no real solutions. Several examples are shown of calculating the discriminant and determining the nature of the roots. The document also discusses how to find the value of a variable if the roots are equal.

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. The Big Nasty 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
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
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
 2 solutions
 1 solution
 0 solutions

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

7. The Discriminant
b2
– 4ac
If b2
– 4ac > 0
If b2
– 4ac = 0
If b2
– 4ac < 0
 2 solutions
 1 solution
 0 solutions
Or
 2 real roots
 1 real root
 no real roots
(equal roots)

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
If b2
– 4ac = 0
If b2
– 4ac < 0
 2 solutions
 1 solution
 0 solutions
( 2 real roots)
(1 real root)
(no real roots)
(equal 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
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
Find the nature of the roots of this equation

12. If equal Roots find value of t.
tx2
+ 8x + 4 = 0
c.f. ax2
+ bx + c = 0
a = t, b = 8,
For equal roots
82
– 4t(4) = 0
64 – 16t = 0
64 = 16t
t = 4
bb22
– 4ac = 0– 4ac = 0
c = 4c = 4
State Rule
Get Values
Sub Values
Solve
Using The Discriminant

13. If equal Roots find value of g.
2x2
– 8x + g = 0
c.f. ax2
+ bx + c = 0
a = 2, b = -8,
For equal roots
(-8)2
– 4(2)g = 0
64 – 8g = 0
64 = 8g
g = 8
bb22
– 4ac = 0– 4ac = 0
c = gc = g
State Rule
Get Values
Sub Values
Solve
Using The Discriminant

14. If equal Roots find value of r.
rx2
– 18x + 27 = 0
c.f. ax2
+ bx + c = 0
a = r, b = -18,
For equal roots
(-18)2
– 4r(27) = 0
324 – 108r = 0
324 = 108r
r = 3
bb22
– 4ac = 0– 4ac = 0
c = 27c = 27
Key Question

15. Equal roots - Nasty
2x2
+ (m+1)x + 8 = 0
c.f. ax2
+ bx + c = 0
a = 2, b = m+1, c = 8
For equal roots
(m+1)2
– 4(2)(8) = 0
m2
+ 2m + 1
m2
+ 2m – 63 = 0
(m + 9)(m – 7) = 0
m = -9 or m = 7
bb22
– 4ac = 0– 4ac = 0
(m+1)(m+1)
=m2
+2m+1
– 64 = 0
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
(-8)2
– 4(2)g < 0
64 – 8g < 0
Inequation
bb22
– 4ac < 0– 4ac < 0

17. Solving Equations
- Reminder
4x – 2 10
4x 12
x 3
=
=
=
Solving Inequations
4x – 2 10
4x 12
x 3
=
>
>
>
(Solution)
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
>