Download Quadratic Equations

Section 1.2
Quadratic Equations
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Quadratic Equations
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Solve the equation:
x  4x  0
2
x( x  4)  0
x  0 or x  4  0
x  0 or x  4
The solution set is 0, 4
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
x2  6  x
Solve the equation:
x  x6  0
2
( x  3)( x  2)  0
x  3 or x  2
The solution set is 3, 2
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Solve each equation.
(a) x  7
2
(b)
 x  3
2
9
x3  9
x 7
x  3  3
x  3  3 or x  3  3
x  7 or x   7

The solution set is  7, 7

x  0 or x  6
The solution set is 0, 6
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Solve by completing the square:
2 x2  6 x  5
2 x2 6 x 5


2
2 2
9 5 9
x  3x   
4 2 4
2
2
3  19

x  
2
4

2x2  6 x  5  0
3
19
x 
2
4
19 3
x

2
2
19  3
19  3
x
or 
2
2
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Use the quadratic formula to find the real solutions if any, of the equation
2x  4x 1  0
2
ax  bx  c  0
2
b2  4ac  (4)2  4(2)(1)  16  8  8
b  4ac  0 so there are two real solutions
which can be found using the quadratic formula.
2
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Use the quadratic formula to find the real solutions if any, of the equation
b  b  4ac
x
2a
2
2x  4x 1  0
2
(4)  8
4 8
x

2(2)
4
2 2

2
 2  2 2  2 
The solution set is 
,

2 
 2
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Use the quadratic formula to find the real solutions if any, of the equation
1 2
x  6 x  18  0
2
x 2  12 x  36  0
ax 2  bx  c  0
b2  4ac  (12)2  4(1)(36)  144  144  0
b  4ac  0 so there is a repeated solution
which can be found using the quadratic formula.
2
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Use the quadratic formula to find the real solutions if any, of the equation
1 2
x  6 x  18  0
2
x  12 x  36  0
2
ax 2  bx  c  0
b  b2  4ac
x
2a
(12)  0
x
2(1)
12

6
2
The solution set is 6
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Use the quadratic formula to find the real solutions if any, of the equation
2x  3  2x
2
Since b  4ac  0,
there is no real solution.
2
2x  2x  3  0
2
ax 2  bx  c  0
b  4ac   2   4(2)(3)
2
2
 4  24  20
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Use the quadratic formula to find the real solutions if any, of the equation
1 2
6  2  0
x x
1  49
x
2(6)
6x2  x  2  0
b  4ac  1  4(6)(2)
2
 1  48  49
2
1  7

12
1  7 1


12
2
1  7
2


12
3
Since b 2  4ac  0,
The solution set is
there are two real solutions.
1 2
 , 
2 3
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
9  x  18   144
2
x  18  4  22 or 14
 x  18
2
 16
x 18  4
22 centimeters by 22 centimeters
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.