Transcript Solving Quadratic Equations
Solving Quadratic Equations Using Square Roots & Completing the Square
Using Square Roots to Solve a Quadratic Equation When a quadratic equation is in the form
ax
2
c
0 Quadratic term and using square roots. 1. 3
x
2 3
x
2
x
2 27 0 27 9
x
2
x x
2 9
x
3 3
or x
3
Solve Using Square Roots 2. 25
x
2 49
x
2 49 25
x
2
x
49 25 7 5
x
7 5
or x
7 5
Other Method:
25
x
2 5
x
49 7 5
x
0 7 0 5
x
7 0
or
5
x
7 0
x
7 5
or x
7 5
Solve Using Square Roots 3. 2
x
2 10 6 2
x
2 16
x
2 8
x
2 2
x
2 8
x
2 2 2.83
Solve Using Square Roots
4. 3
x
2
18
2 3
x
2
20
x x
2 2
20 3 20 3
x
20 3
x
2 5 3
x
2 15 3 3 3
Solve Using Square Roots
5.
x
2
20
11
x
2
9
The square root of a negative number is undefined in the set of real numbers.
x x
2
9
x has no real solution
Solve Using Square Roots 6.
x
2 12 12
x
2 0
x x
2
x
0 0 0
Solve Using Square Roots
7. 4( (
x x
6)
2
6)
2
8
x
6
2 2 32 (
x
6)
2
x
6
2 2 8
x
6
2 2
x
6
2 2
x
3.17
x
8.83
or or x
6
2 2
8.
Solve Using Square Roots
Perfect Square Trinomial
x
2 12
x
36 64 (
x
6) 2 64
Binomial Square
(
x
6) 2 64
x x
6 6 8
x
14 8
or x
6 8
or x
2
Can we solve this quadratic using a different method?
Solve by Factoring 8
b
.
x
2 12
x
36 64
x
2 12
x
36 64 0
x
2 12
x
28 0 (
x
14)(
x
2) 0
x
14
x
0 14
or x
2 0
or x
2
9.
x
2 Solve by Factoring 8
x
9 0
This trinomial does NOT factor
!!!
We will Use a method called Completing the Square to solve.
Solve by Completing the Square NOT a Perfect Square Trinomial 9.
x
2 8
x
9 0
x
2 8
x
__ 2 9 __ 2
x
2 8
x
16 9 16 (
x
4) 2 7
1. Move the constant to the other side.
2. Add the square of half the linear coefficient to both sides.
3. Factor the perfect square trinomial into a binomial square.
9.
continued
(
x
4) 2 7
x
4 2 7
x
4
x
4 7 7
or x
4
x
4 7 7
Solve by Completing the Square NOT a Perfect Square Trinomial 10.
x
2 4
x
8 0
x x
2 2 (
x
4 4
x
__ 2 8 __ 2 2)
x
2 4 8 12 4
1. Move the constant to the other side.
2. Add the square of half the linear coefficient to both sides.
3. Factor the perfect square trinomial into a binomial square.
10.
continued
(
x
2) 2 12
x
2 2 12
x
2
x
2 2 3
x
2 2 3 2 3
or x
2 2 3
x
1.46
or x
5.46
11.
x
2 Solve by Completing the Square 3
x
7 NOT a Perfect Square Trinomial
x
2 3
x
3 __ 2 7 3 2 __
x
2 3
x
9 4 7
x
3 2 2 28 4 9 4 9 4
1. Move the constant to the other side.
2. Add the square of half the linear coefficient to both sides.
3. Factor the perfect square trinomial into a binomial square.
11.
continued
x
3 2 2
x
3 2 2 28 4 9 4 37 4
x
3 2
x
3 2 37 2
or x
37 2 2 37 2
x
3 2 37 2
x
3 2 37
12. 4
x
2 Solve by Completing the Square 4 4
x
7 0 4
x
2
x
7 0 4
x
2
x
1 2 2 ___
x
2
x
x
1 2 2 1 4 7 7 4 4 1 4 8 4 1 2 2
Divide both sides by leading coefficient.
1. Move the constant to the other side.
2. Add the square of half the linear coefficient to both sides.
3. Factor the perfect square trinomial into a binomial square.
12.
continued
x
1 2 2 2
x
1 2 2 2
x
1 2
x
1 2 2 2
or x
2 2
x
1 2 2
x x
1 2 2 2 2 1 2 2 2