Post on 05-Jan-2016
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”
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
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
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
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
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
COMPLETE THE SQUARE
x2 + 12x + _____ = 3 + _____36 36
COMPLETE THE SQUARE
x2 – 8x + _____ = 10 + _____16 16
COMPLETE THE SQUARE
x2 – 20x + _____ = 1 + _____100 100
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
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
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
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.
FLASH CARDS
In the equation, x2 + 5x – 6 = 0
a = 1
FLASH CARDS
In the equation, x2 + 5x – 6 = 0
b = 5
FLASH CARDS
In the equation, x2 + 5x – 6 = 0
c = -6
FLASH CARDS
In the equation, x2 + 5x – 6 = 0
the discriminant =
49
FLASH CARDS
In the equation, 3x2 – 6 = 0
the discriminant =
72
FLASH CARDS
How many roots if thediscriminant is equal to
120
Two real roots
FLASH CARDS
How many roots if thediscriminant is equal to
0
A double root
FLASH CARDS
How many roots if thediscriminant is equal to
13
Two real roots
FLASH CARDS
How many roots if thediscriminant is equal to
-15
No real roots