Completing the Square
To solve
Step 1 If divide each side by a.
Step 2 Write the equation with the variable terms on
one side and the constant on the other.
Step 3 Take half the coefficient of xand square it.
Step 4 Add the square to each side.
Step 5 Factor the perfect square trinomial, and write it
as the square of a binomial. Simplify the other
side.
Step 6 Use the square root property to complete the
solution.
aZ1,ax^2 +bx+c= 0 1 with aZ 02 :CONCEPTS EXAMPLES
Solve
Divide by 2.
Add 9.Add 1.
Factor. Add.
or Square root
orpropertyThe solution set is E 1 + 210 , 1- (^210) F,or E 1 (^210) F
x= 1 + 210 x = 1 - 210
x- 1 = 210 x - 1 =- 210
1 x- 122 = 10
x^2 - 2 x+ 1 = 9 + 1
C
1
2 1 -^22 D(^2) = 1 - 122 = 1
x^2 - 2 x= 9
x^2 - 2 x- 9 = 0
2 x^2 - 4 x- 18 =0.
9.2 The Quadratic Formula
Quadratic Formula
The solutions of are
given by
The Discriminant
If a, b, and c are integers, then the discriminant,
of determines the number
and type of solutions as follows.
b^2 4 ac, ax^2 +bx+c= 0
xb 2 b^2 4 ac
2 a.
ax^2 +bx+c= 0 1 with aZ 02SolveorThe solution set isFor the discriminant is
Two rational solutions
For the discriminant is
12 - 4142112 =- 15. Two nonreal complex solutions4 x^2 +x+ 1 =0,32 - 41121 - 102 = 49.
x^2 + 3 x- 10 =0,E-1, -
2
3 F.x=- 5 - 1
6
x= =- 1- 5 + 1
6
=-
2
3
x=- 5 252 - 4132122
2132
=
- 5 1
6
3 x^2 + 5 x+ 2 =0.Number and
Discriminant Type of Solutions
Positive, the square Two rational solutions
of an integer
Positive, not the Two irrational solutions
square of an integer
Zero One rational solution
Negative Two nonreal complex solutions9.3 Equations Quadratic in Form
A nonquadratic equation that can be written in the form
for and an algebraic expression u, is called
quadratic in form. Substitute ufor the expression, solve
for u, and then solve for the variable in the expression.
aZ 0au^2 buc0,Solve
Let
Factor.ororor Subtract 5.The solution set is E-7, -^163 F.x=- x=- 716
3
x + 5 =- x+ 5 =- 2 x+ 5 =u1
3
u=- u=- 21
3
13 u+ 121 u+ 22 = 03 u^2 + 7 u+ 2 = 0 u=x+5.31 x+ 522 + 71 x+ 52 + 2 =0.560 CHAPTER 9 Quadratic Equations, Inequalities, and Functions
(continued)