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
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 s
re
oa
lu
l trioo
nt
s
s
1 s
re
oa
lu
l trioo
nt(equal roots
n0osoreluatlioronosts
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 - Nasty
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
