Transcript Document

Math 20-1 Chapter 4 Quadratic Equations
4.4 Quadratic Formula
Teacher Notes
4.3 Quadratic Formula
The roots of a quadratic equation are…
• the solutions for the variable.
• related to the zeros of the corresponding function.
• related to the x-intercepts of the graph of the corresponding
function.
No real number x-intercepts
No real number zeros
One real x-intercept
One real zeros
Two real x-intercepts
Two real zeros
No real number roots
…solutions
Two equal real roots
One real solution
Two distinct real roots
Two real solutions
4.3.1
The Quadratic Formula
The roots of the quadratic equation
ax2 + bx + c = 0 can be found by using the
quadratic formula:
b  b  4ac
x
2a
2
4.3.2
Deriving The Quadratic Formula
Determine the solutions by completing the square.
ax  bx  c  0
ax2  bx  c
 2 b 
a  x  x   c
a 

c
b2 
b2
 2 b
 x  x  4a 2    4a 2 
a
a


2
b  b  4ac

x
 
2
2
a
4
a


2
b
b  4ac
x 
2a
4a 2
2
2
b
b2  4ac
x 
2a
4a 2
b
b 2  4ac
x 
2a
4a 2
b
b  4ac
x 
2a
2a
2
b  b  4ac
x
2a
2
4.3.3
Two Equal Real Roots
ax 2  bx  c  0
Solve x2 + 3x - 2 = 0.
x = – b + b2 – 4ac
2a
x = – 3 + 32 – 4(1)(–2)
2(1)
Quadratic formula
a = 1, b = 3, c = –2
x = – 3 + 17
2
The solutions are x = – 3 + 17
2
x = – 3 – 17
– 3.56.
2
Simplify. PEMDAS
Two distinct real roots
0.56 or
Graph y = x2 + 3x – 2 and note
CHECK
that the x-intercepts are
approx. 0.56 and – 3.56.
4.3.4
Solving Quadratic Equations Using the Quadratic Formula
Solve 2x2 - 5x + 2 = 0.
a = 2, b = -5, c = 2
b  b 2  4ac
x
2a
x
(5)  (5)2  4(2)(2)
x
2(2)
8
2
x  or x 
4
4
5  25  16
x
4
1
x  2 or x 
2
5 9
x
4
53
x
4
53
5 3
or x =
4
4
Two distinct real roots
4.3.5
Solving Quadratic Equations that have No Real Roots
Solve x2 - 5x + 7 = 0.
b  b 2  4ac
x
2a
(5)  (5)  4(1)(7)
x
2(1)
2
5  3
x
2
No real roots
4.3.6
Solving Quadratic Equations with Two Equal Real Roots
Solve x2 - 6x + 9 = 0.
(6)  (6) 2  4(1)(9)
x
2(1)
6 0
x
2
Two Equal real roots
6
x
2
x3
4.3.7
Page 254
#3a,c,e, 4, 6, 10, 11, 17, 21
4.3.12
Determining The Nature of the Roots
http://www.udidahan.com/2009/06/29/dont-create-aggregate-roots/
http://80sbabiesthink.wordpress.com/2008/06/23/roots/
Determining The Nature of the Roots
The quadratic formula
b  b2  4ac
x
2a
will give the roots of the quadratic equation.
From the quadratic formula, the radicand,
b2 - 4ac, will determine the Nature of the Roots.
By the nature of the roots, we mean:
• whether the equation has real roots or imaginary
• if there are real roots, whether they are different or
equal
The radicand b2 - 4ac is called the discriminant
of the equation ax2 + bx + c = 0 because it discriminates
among the three cases that can occur.
4.3.8
The discriminant describes the Nature of the Roots
of a Quadratic Equation
b  b2  4ac
x
2a
If b2 - 4ac > 0, then there are
two different real roots.
If b2 - 4ac = 0, then there are
two equal real roots.
If b2 - 4ac < 0, then there are no
real roots.
4.3.9
Use the discriminant to determine the nature of the roots.
b  b2  4ac
x
2a
Equation
ax2 + bx + c = 0
Discriminant
b2 – 4ac
a.
2x2 + 6x + 5 = 0
62 – 4(2)(5) = –4
b.
x2 – 7 = 0
c.
4x2 – 12x + 9 = 0 (–12)2 –4(4)(9) = 0
02 – 4(1)(– 7) = 28
Nature of
Roots
No real roots
Two distinct
real roots
Two equal real
roots
4.3.10
Determine the value of k for which the equation
x2 + kx + 4 = 0 has
a) equal roots b) two distinct real roots c) no real roots
a) For equal roots, b2 - 4ac = 0.
Therefore, k2 - 4(1)(4) = 0
The equation has equal roots
when k = 4 and k = -4.
k2 - 16 = 0
k2 = 16
k=+4
b) For two different real roots, b2 - 4ac > 0.
k2 - 16 > 0
k2 > 16
Therefore, k > 4 or k < -4. This may be written as | k | > 4.
c) For no real roots, b2 - 4ac < 0
k2 < 16
Therefore, -4 < k < 4. This may be written as | k | < 4.
4.3.11
Assignment:
P. 254
1a,b,c, 2d,e, 14, 17, 21