SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE,...
-
Upload
angelica-elliott -
Category
Documents
-
view
213 -
download
0
Transcript of SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE,...
![Page 1: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/1.jpg)
SOLVING QUADRATIC EQUATIONSBY COMPLETING THE SQUARE
BECAUSE GRAPHING IS SOMETIMESINACCURATE, ALGEBRA CAN BE
USED TO FIND EXACT SOLUTIONS.
ONE OF THOSEALGEBRAIC METHODS IS
“COMPLETING THE SQUARE”
![Page 2: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/2.jpg)
SOLVING QUADRATIC EQUATIONSBY COMPLETING THE SQUARE
Let’s solvex2 – 10x + 18 = 0
Step 1:Get rid of constant on the left side
x2 – 10x + 18 = 0-18 -18
x2 – 10x = -18
![Page 3: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/3.jpg)
SOLVING QUADRATIC EQUATIONSBY COMPLETING THE SQUARE
Let’s solvex2 – 10x + 18 = 0
Step 2:Add constant to left side to create PST
Half of middle term, then square it.
x2 – 10x = -18+ 25
Must add it to BOTH sides.
+ 25
(x – 5)2 = 7
![Page 4: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/4.jpg)
SOLVING QUADRATIC EQUATIONSBY COMPLETING THE SQUARE
Let’s solvex2 – 10x + 18 = 0
Step 3:Square root of both sides.
(x – 5)2 = 7
75 x
![Page 5: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/5.jpg)
SOLVING QUADRATIC EQUATIONSBY COMPLETING THE SQUARE
Let’s solvex2 – 10x + 18 = 0
Step 4:Solve left side for x
(x – 5)2 = 7
75 x+5 +5
75x
![Page 6: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/6.jpg)
SOLVING QUADRATIC EQUATIONSBY COMPLETING THE SQUARE
Try this onex2 + 6x – 3 = 0
(x + 3)2 = 12
123 x-3 -3
123x
x2 + 6x = 3+3 +3
Half of 6,squared
+9 +9
x2 + 6x + 9 = 12
![Page 7: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/7.jpg)
COMPLETE THE SQUARE
x2 + 12x + _____ = 3 + _____36 36
![Page 8: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/8.jpg)
COMPLETE THE SQUARE
x2 – 8x + _____ = 10 + _____16 16
![Page 9: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/9.jpg)
COMPLETE THE SQUARE
x2 – 20x + _____ = 1 + _____100 100
![Page 10: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/10.jpg)
SOLVING QUADRATIC EQUATIONSUSING THE QUADRATIC FORMULA
Standard form for quadratic equations is
ax2 + bx + c = 0
and can be solved using theQuadratic Formula:
a
acbbx
2
42
![Page 11: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/11.jpg)
SOLVING QUADRATIC EQUATIONSUSING THE QUADRATIC FORMULA
Example: 3x2 + 7x – 2 = 0
a
acbbx
2
42
)3(2
)2)(3(477 2
6
24497
6
737
![Page 12: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/12.jpg)
THE DISCRIMINANT
In a quadratic formula, the discriminant is the expression
under the racical sign.
a
acbbx
2
42
What is the discriminant for 4x2 + 2x – 7 = 0 ?
b2 – 4ac = 22 – 4(4)(-7) = 4 + 112 = 116
![Page 13: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/13.jpg)
THE DISCRIMINANT
The discriminant tells you something about the roots of the equation.
If the discriminant is negative, (b2 – 4ac < 0), then there are no real roots (no solutions).
If the discriminant is zero, (b2 – 4ac = 0), then there is a double root (one solution).
If the discriminant is positive, (b2 – 4ac > 0), then there are two real roots.
![Page 14: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/14.jpg)
FLASH CARDS
In the equation, x2 + 5x – 6 = 0
a = 1
![Page 15: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/15.jpg)
FLASH CARDS
In the equation, x2 + 5x – 6 = 0
b = 5
![Page 16: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/16.jpg)
FLASH CARDS
In the equation, x2 + 5x – 6 = 0
c = -6
![Page 17: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/17.jpg)
FLASH CARDS
In the equation, x2 + 5x – 6 = 0
the discriminant =
49
![Page 18: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/18.jpg)
FLASH CARDS
In the equation, 3x2 – 6 = 0
the discriminant =
72
![Page 19: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/19.jpg)
FLASH CARDS
How many roots if thediscriminant is equal to
120
Two real roots
![Page 20: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/20.jpg)
FLASH CARDS
How many roots if thediscriminant is equal to
0
A double root
![Page 21: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/21.jpg)
FLASH CARDS
How many roots if thediscriminant is equal to
13
Two real roots
![Page 22: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.](https://reader036.fdocuments.us/reader036/viewer/2022081519/56649f2b5503460f94c46092/html5/thumbnails/22.jpg)
FLASH CARDS
How many roots if thediscriminant is equal to
-15
No real roots