simplify – do not use a calculator 1) √24 2) √80 1) √24 2) √80
TRANSCRIPT
Simplify – Do not use a calculator
1) √24
2) √80
4.6 Solving Quadratic Equations by
Completing the Square
Learning Target: I can solve equations by completing the
square
Perfect Square Trinomials
Examples x2 + 6x + 9 x2 - 10x + 25 x2 + 12x + 36
2( 5) 64x 5 8x
When you take the square root, You MUST consider the Positive and Negative answers.
5 8x 5 8x 5 5
13x 5 5
3x
PerfectSquare
On One side
Take Square Root
ofBOTH SIDES
2 ( 5) 64x
PerfectSquare
On One side
Take Square Root
ofBOTH SIDES
But what happens if you DON’T have a perfect square on one side…….
You make it a Perfect Square
Use the relations on next slide…
2( 6)x ( 2 ) To expand a perfect square binomial:
2 12 36x x 6x 26
We can use this relationship to find the missing term….To make it a perfect square trinomial that can be factored into a perfect square binomial.
2 _ _12 _x x 12 2 6 626 36
36
2x
Take ½ middle term
Then square it
The resulting trinomial is called a perfect square trinomial,
which can be factored into a perfect square binomial.
2 _ _18 _ _x x
18 2 92(9) 81
81 2( 9)x
1. 2 12 0x x
1. Make one side a perfect square
2. Add a blank to both sides
3. Divide “b” by 2
4. Square that answer.
5. Add it to both sides
6. Factor 1st side
7. Square root both sides
8. Solve for x
2 0x x ___ ___12 2 6
2(6) 36
36 362( 6)x 362( 6) 36x 6 6x
6 6x 6 6x 6 6
12x 6 6
0x
12
Perfect Square Trinomials
Create perfect square trinomials.
x2 + 20x + ___ x2 - 4x + ___ x2 + 5x + ___
100
4
25/4
Steps to solve by completing the square
1.) If the quadratic does not factor, move theconstant to the other side of the equation Ex: x²-4x -7 =0 x²-4x=7
2.) Work with the x²+ x side of the equation and complete the square by taking ½ of the coefficientof x and squaring Ex. x² -4x 4/2= 2²=4
3.) Add the number you got to complete the square toboth sides of the equationEx: x² -4x +4 = 7 +4
4.)Simplify your trinomial square Ex: (x-2)² =11
5.)Take the square root of both sides of the equationEx: x-2 =±√11
6.) Solve for xEx: x=2±√11
Solving Quadratic Equations by Completing the Square
Solve the following equation by completing the square:
Step 1: Set quadratic equation equal to zero 2 8 20 0x x
2 8 20x x
Solving Quadratic Equations by Completing the Square
Step 2: Find the term that completes the square. Add that term that is equal to zero into the equation.
X2 + 8x + ____ + _____ -20 = 016 -16
Solving Quadratic Equations by Completing the Square
Step 3: Factor the terms that create the perfect square trinomial. Simplify the other 2 terms of the equation.
X2 + 8x + ____ + _____ -20 = 016 -16
(x + 4)(x + 4) - 36 = 0
(x + 4)2 - 36 = 0
Note: This is vertex form of the equationy =(x + 4)2 - 36
Solving Quadratic Equations by Completing the Square
Step 4: Move the constant term and isolate the square binomial.
(x + 4)2 - 36 = 0 (x + 4)2 = 36
Solving Quadratic Equations by Completing the Square
Step 5: Take the square root of each side
2( 4) 36x
( 4) 6x
Solving Quadratic Equations by Completing the Square
Step 6: Set up the two possibilities and solve
4 6
4 6 an
d 4 6
10 and 2 x=
x
x x
x
Was there an easier way?2 8 20x x
Solve by Completing the Square2 6 16 0x x
2 6 16x x +9 +9
2 6 9 25x x 2
3 25x 3 5x
3 5x 8x 2x
Solve by Completing the Square
2 22 21 0x x 2 22 21x x
+121 +1212 22 121 100x x
211 100x
11 10x 11 10x 21x 1x
Solve by Completing the Square
2 2 5 0x x 2 2 5x x
+1 +12 2 1 6x x
21 6x
1 6x 1 6x
Solve by Completing the Square
2 10 4 0x x 2 10 4x x
+25
+252 10 25 29x x
25 29x 5 29x
5 29x
Solve by Completing the Square
01182 xx1182 xx
+16
+16 51682 xx
54 2 x 54 x
54 x
Solve by Completing the Square
0462 xx462 xx
+9 +9
5962 xx 53 2 x 53 x
53x
Assignment pg 237-238
Homework– p. 237 1-11
Challenge - 76
Completing the Square-Example #2
Solve the following equation by completing the square:
Step 1: If the lead coefficient is not 1, factor the lead coefficient from the a and b terms.
2x2 + 12x - 5 = 0
2(x2 + 6x) - 5 = 0
Solving Quadratic Equations by Completing the Square
Step 2: Find the term that completes the square.
(remember add in zero)
The quadratic coefficient must be equal to 1 before you complete the square, so you must divide the first 2 terms by the quadratic coefficient first.
2(x2 + 6x + ___) +___ - 5 = 09
Solving Quadratic Equations by Completing the Square
Step 2: What makes out of the parenthesis zero?
2(x2 + 6x + ___) +___ - 5 = 09 -18
Solving Quadratic Equations by Completing the Square
Step 3: Factor the perfect square trinomial in the equation. Combine the other two terms.
2(x2 + 6x + ___) +___ - 5 = 0
2(x + 3)2 - 23 = 0
9 -18
Solving Quadratic Equations by Completing the Square
Step 4: Move the constant to the right side of the equation and solve.
Isolate the perfect square
Take the square root of both sides
Solving Quadratic Equations by Completing the Square
Step 4 continued:
Solving Quadratic Equations by Completing the Square
Try the following examples. Do your work on your paper and then check your answers.
1. x2 + 2x - 63 = 0
2. x2 - 10x - 15 = 0
3. 2x2 - 6x - 1 = 0