Post on 22-Dec-2015
Solving Quadratic Equations Using the Quadratic Formula11.211.2
1. Solve quadratic equations using the quadratic formula.2. Use the discriminant to determine the number of real
solutions that a quadratic equation has.3. Find the x- and y-intercepts of a quadratic function.4. Solve applications using the quadratic formula.
Solve:Solve:
ac
xab
x 2
2x
02 cbxax
½ squared
Coefficient of squared term is NOT 1.
02 ac
xab
x
2
2
4
4
a
acb
2
22
4
42 a
acbab
x
aacb
ab
x2
42
2
ab
ab
221
2
2
4a
b
ab2
2
2
4a
b22
2
4
4
4 a
ac
a
b
aacbb
x2
42
aacb
ab
x2
42
2 Quadratic FormulaQuadratic Formula
Quadratic Formula
To solve ax2 + bx + c = 0, where a 0, use
2 4
2
b b acx
a
2 4
2
b b acx
a
a = 2
27 7 2 34
22x
7 49 24
4x
7 25
4x
7 5
4x
7 5
4x
7 5
4x
3x 1
2x
Solve:Solve: 372 2 xx
0372 2 xx
3,
21
2 real rational solutions
, b = –7 , c = 3
2 2 5 6x x
22 2 4 1 11
2 1x
2 4 44
2x
2 48
2x
2 2 11 0x x
a = 1, b = –2, c = –11.
2 4 3
2 2x
1 2 3x
Solve:Solve:
321,321
2 real irrational solutions
16 ∙3
2 4 3
2x
1
1
2
224 i
x
Solve:Solve:
2 non-real complex solutions
xx 452
0542 xxa = 1, b = -4, c = 5
12
51444 2 x
244 x
ii 2,2
ix 2
2 1
1
Slide 11- 7Copyright © 2011 Pearson Education, Inc.
Solve using the quadratic formula.
a)
b)
c)
d)
23 8 2 0x x
8 2 10
6
8 10
3
4 10
3
2 22
3
11.2
Slide 11- 8Copyright © 2011 Pearson Education, Inc.
Solve using the quadratic formula.
a)
b)
c)
d)
23 8 2 0x x
8 2 10
6
8 10
3
4 10
3
2 22
3
11.2
Method When the Method is Beneficial
1. Factoring Use when the quadratic equation can be easily factored.
2. Square root principle
Use when the quadratic equation can be easily written in the form
No middle term.
3. Completing the square
Rarely the best method, but important for other topics.
4. Quadratic formula
Use when factoring is not easy, or possible.
Methods for Solving Quadratic Equations
2 2, or ( ) .ax c ax b c
7 25
4x
0372 2 xx
3,
21
2 real rational solutions
2 48
2x
2 2 11 0x x
321,321
2 real irrational solutions
2 non-real complex solutions
0542 xx
244 x
ii 2,2
What made the difference? The DiscriminantThe Discriminant
acb 42
Discriminant:The discriminant is the radicand, b2 – 4ac, in the quadratic formula.
The discriminant is used to determine the number and type of solutions to a quadratic equation.
7 25
4x
0372 2 xx
3,
21
2 real rational solutions
2 48
2x
2 2 11 0x x
321,321
2 real irrational solutions
2 non-real complex solutions
0542 xx
244 x
ii 2,2
If the discriminant is…. there will be….
positive and a perfect square 2 real rational solutions. There will be no radicals left in the answer. The equation could have been factored.
positive but not a perfect square 2 real irrational solutions. There will be a radical in the answer.
0 1 real rational solution.
negative 2 non-real complex solutions. The answer will contain an imaginary number.
Use the discriminant to determine the number and type of solutions. Use the discriminant to determine the number and type of solutions.
22 5 1x x
22 5 1 0x x
Evaluate the discriminant: b2 – 4ac.
25 4 2 1
a = 2, b = 5, c = 1
25 8
17
Two real irrational solutions.
Positive but not a perfect square.
Discriminant:
Slide 11- 14Copyright © 2011 Pearson Education, Inc.
Find the discriminant.
a) 5
b) 73
c) 25
d)
2 7 6x x
11.2
73
Slide 11- 15Copyright © 2011 Pearson Education, Inc.
Find the discriminant.
a) 5
b) 73
c) 25
d)
2 7 6x x
11.2
73
Slide 11- 16Copyright © 2011 Pearson Education, Inc.
Determine the number and type of solutions.
a) Two real rational solutions
b) Two real irrational solutions
c) One real rational solution.
d) Two non-real complex solutions.
2 7 6x x
11.2
Slide 11- 17Copyright © 2011 Pearson Education, Inc.
Determine the number and type of solutions.
a) Two real rational solutions
b) Two real irrational solutions
c) One real rational solution.
d) Two non-real complex solutions.
2 7 6x x
11.2
Slide 11- 18Copyright © 2011 Pearson Education, Inc.
Find the discriminant.
a)
b) 136
c) -104
d)
11.2
104
22 4 15x x
136
Slide 11- 19Copyright © 2011 Pearson Education, Inc.
Find the discriminant.
a)
b) 136
c) -104
d)
11.2
104
22 4 15x x
136
Slide 11- 20Copyright © 2011 Pearson Education, Inc.
Determine the number and type of solutions.
a) Two real rational solutions
b) Two real irrational solutions
c) One real rational solution.
d) Two non-real complex solutions.
11.2
22 4 15x x
Slide 11- 21Copyright © 2011 Pearson Education, Inc.
Determine the number and type of solutions.
a) Two real rational solutions
b) Two real irrational solutions
c) One real rational solution.
d) Two non-real complex solutions.
11.2
22 4 15x x
2xxf
122 xxxf
122 xxy
120 2 xx
What are we finding?
x-intercepts
34
340
xx
xx
12,0
y-intercept
1212000 2 f
0,30,4