Chapter 4: Polynomial and
Rational Functions
Section 4-2: Quadratic Equations
Solving Quadratic Equations –
Finding the zeros
There are 4 ways to solve a quadratic. You should be
familiar with all of them from Algebra 2.
Graphing: Can be difficult to graph accurately.
Factoring: This is the easiest and best method. But
can be used only IF the equation is factorable.
Completing the Square: Works for any quadratic.
Quadratic Formula: Works for any quadratic
Standard Form of Quadratic
•The standard form of a quadratic equation is
Ax2+Bx+c=0
•The discriminant is very useful to tell us how
many real roots a quadratic equation has.
•
B 2 − 4 AC > 0 Two distinct real roots/zeros
•
B 2 − 4 AC = 0
Exactly one real root/zero
•
B 2 − 4 AC < 0
No real roots/zeros (imaginary)
Proving the quadratic formula
•Given a quadratic equation in standard form,
2 + bx + c = 0
prove that
ax
2
x = −
b ±
b −4 ac
2a
bx c
+ =0
a
a
bx
c
x2 +
=−
a
a
x2 +
b 2
c
b2
) =− +
(x +
2a
a 4a2
b 2 b2 − 4ac
) =
(x +
2a
4a2
b
=±
x+
2a
b2 − 4ac
2a
b ± b2 − 4ac
x=−
2a
Imaginary roots
• Imaginary roots of polynomial
equations with real coefficients always
occur in conjugate pairs.
• A pair of complex numbers in the form
• a+bi and a-bi are called conjugates.
Example #1
•Find the zeros of
4a 2 + 6a − 3 = 0 by completing the square.
3
3
a2 + a =
2
4
3
3 9
(a + ) 2 = +
4
4 16
3 2 21
(a + ) =
4
16
3
21
a+ =±
4
4
3
21
a=− ±
4
4
− 3 ± 21
a=
4
Example #2
•Find the discriminant of X2+2x-2=0 and
describe the nature of the roots of the
equation. Then solve the equation by using
the Quadratic Formula.
a=1
b=2 The discriminant is 4-4(-2)=12 so there are two distinct
c=-2 real roots.
− 2 ± 4 − 4(−2)
x=
2
− 2 ± 12
x=
2
−2±2 3
x=
2
x = −1 ± 3
HW#26
•
•
•
•
Section 4-2
Pp 219-221
#5,12,15,16,20,21,22,25,
26,31,32,35,40,41
Download

Chapter 4: Polynomial and Rational Functions Section 4-2: Quadratic Equations