Section 5-4 Completing The Square & Quadratic … · Notice – 5 2 it is the number that when...
8
Notice – 2 5 it is the number that when added to itself is equal to –5 and when multiplied by itself is equal to 4 25 . Section 5-4 Completing The Square & Quadratic Formula EASY 0 6 10 2 x x 0 6 10 2 x x Move constant to the other side 6 10 2 x x 25 6 25 10 2 x x 31 25 10 2 x x Easily Factors 31 5 5 x x Can be re-written 31 5 2 x Take the square root of both sides 31 5 2 x Don’t forget . 31 5 x Isolate x 31 5 x OR 5678 . 10 5678 . 0 or x MEDIUM 0 12 15 3 2 x x 0 9 15 3 2 x x Move constant to the other side 9 15 3 2 x x Divide both sides by the leading coefficients 3 9 3 15 3 2 x x 3 5 2 x x 4 25 1 3 4 25 10 2 x x 4 25 4 12 2 5 2 5 x x 4 13 2 5 2 x 4 13 2 5 2 x 2 13 2 5 x 2 13 2 5 x 2 13 5 x 3028 . 4 6972 . 0 or x +6 +6 5 ½ of 10 25 Square It –5 –5 –9 –9 2 5 ½ of 5 Square 5/2 Add 25 to both sides Add 25 to both sides 4 25
Transcript of Section 5-4 Completing The Square & Quadratic … · Notice – 5 2 it is the number that when...
Notice – 25 it is the
number that when added to itself is equal to –5 and when multiplied by
itself is equal to 425 .
Section 5-4 Completing The Square & Quadratic Formula
EASY 06102 xx 06102 xx Move constant to the other side
6102 xx 25625102 xx
3125102 xx Easily Factors
3155 xx Can be re-written
315 2 x Take the square root of both sides
315 2 x Don’t forget .
315 x Isolate x
315x OR 5678.105678.0 orx
MEDIUM 012153 2 xx 09153 2 xx Move constant to the other side
9153 2 xx Divide both sides by the leading coefficients
39
3153 2
xx
352 xx 425
13
425102 xx
425
412
25
25
xx 4
1325 2
x
413
25 2
x 2
1325 x 2
1325 x 2
135x
3028.46972.0 orx
+6 +6
5 ½ of
10 25 Square
It
–5 –5
–9 –9
25
½ of 5
Square 5/2
Add 25 to both sides Add 25 to both sides
425
Solve the following by completing the square.
1. 019102 xx 2. 02122 xx
3. 08152 xx 4. 0472 xx
5. 01482 2 xx 6. 05183 2 xx
Solve the following by completing the square.
7. 03102 2 xx 8. 05213 2 xx
9. 0852 2 xx 10. 07103 2 xx
Solve the following by completing the square. Derive the quadratic formula.
02 cxbxa
aacbbx
242
The Quadratic Formula Solve the following by quadratic formula
1. 019102 xx 2. 02122 xx
3. 08152 xx 4. 0472 xx
5. 01482 2 xx 6. 05183 2 xx
7. 03102 2 xx 8. 05213 2 xx
aacbbx
242
The DISCRIMINANT The discriminant determines the type of solution Rational or Irrational and Real or Imaginary.
aacbbx
242
acb 42
Describe the nature of the roots using the discriminant. Then, use the quadratic equation to solve.
1. 01482 2 xx 2. 0823 2 xx
3. 012153 2 xx 4. 07103 2 xx
5. 018248 2 xx 6. 01362 xx
7. 0243412 2 xx 8.
Positive Perfect Square
Examples = 9, 25, 49
If the discriminate is a positive perfect square then the square root can be completely eliminated when the radical is simplified and no radical will be left over. So the solution can be written as 2 REAL and RATIONAL number.
Positive Non Square
Examples = 7, 12, 22
If the discriminate is positive but NOT a perfect square then there will still be 2 REAL solutions but the solutions will be IRRATIONAL since the radical cannot be completely eliminated.
Zero
Examples = 0
If the discriminate is zero then there will be just 1 REAL RATIONAL solution because adding or subtracting 0 is equivalent.
Negative
Examples = – 3, – 9, – 12
If the discriminate is negative then there will be 2 IMAGINARY solutions (there will be an i in the solution).
acb 42 = discriminant
Give a possible value for ‘a’ such that the solution of the quadratic would have 2 real rational solutions.
0252 xax
Applications
1. Find the value of x that would make the diagram below accurate.
2. A golf ball is hit with an initial vertical velocity of 80 fps 216 80h t t
a. How high is the ball after 2 seconds?
b. How many seconds would it take the ball to hit the ground (the height would be h=0)?
c. When will the ball reach 48 feet?
d. When does the ball reach a maximum height?
e. What is the average height from t = 1 to t = 2 seconds?
f. For what values of t is the domain appropriate?
x+2
x x+4
3. A baseball his hit with an initial vertical velocity of 121 fps an the ball was struck 1 foot above ground.
112116 2 tth
a. How high is the ball after 2 seconds?
b. How many seconds would it take the ball to hit the ground (the height would be h=0)?
c. When will the ball reach 30 feet?
d. When does the ball reach a maximum height?
e. What is the average height from t = 0.2 to t = 0.9 seconds?
4. A person at a framing store is making a frame mat to go around a picture. The mat is a uniform 2 inches around on each side. The picture’s width is 5 less than twice the picture’s height. The entire area of the frame with picture included is 221 square inches. What are the dimensions of the picture?
