Transcript Solution

Multiple Methods for
Solving Quadratics
Section P.5
Definition: Quadratic Equation in x
A quadratic equation in x is one that can be written
in the form
ax 2 + bx + c = 0
where a, b, and c are real numbers with a = 0.
 Now, our five methods of solving for today…
Method #1: Factoring
1. Set the equation equal to zero
2. Completely factor the quadratic equation
3. Set each factored part separately equal to zero,
solve for the unknown in each
Guided Practice:
2
Solve for x: x – 6x + 5 = 0
Factored form: (x – 5)(x – 1) = 0
x–5=0
OR
x–1=0
Zero Factor Property
x=5
OR
x=1
Guided Practice:
2
Solve by factoring: x – 5x = 14
x = –2, 7
Guided Practice:
3
8
–
,
Solve by factoring: 6x – 7x – 24 = 0 x =
2
3
2
Hint: use the “grouping method”
Method #2: Graphically
1. Graph the quadratic equation (set an appropriate
viewing window)
2. Calculate the zeros (x-intercepts) using your
grapher
3. Note: if you use your graph, you must always
include a sketch of that graph with your solution!
Back to the first problem…
2
Solve for x: x – 6x + 5 = 0
This time, use your grapher!!!
Method #3: Extracting Square Roots
1. Get the “squared” term by itself on one side of
the equation
2. Take the square root of both sides (remember to
take either a positive or negative answer when
“extracting” the roots!)
3. Solve for the unknown (two separate equations)
Guided Practice:
2
Solve by extracting square roots: (2x – 1) = 9
Solution: x = 2, –1
Guided Practice:
2
Solve by extracting square roots: 6w – 13 = 15 – 3w 2
+
Solution: w = –
2 7
3
Method #4: Completing the Square
1. Collect the “x” terms by themselves on one side
of the equation
2. Factor the “x” terms so that the x 2 coefficient is 1
3. Add (b/2)2 to both sides of the equation
4. Factor the new “x” terms in the equation
5. Solve for x by extracting roots, as in the previous
method
Guided Practice:
2
Solve by completing the square: 4x – 20x + 17 = 0
5
Solution: x   2
2
So, what is this
“most famous
formula?”…
(method 5 by the way…)
I’m speaking, of course, of the
Quadratic Formula:
2
The solutions of the quadratic equation ax + bx + c = 0,
where a = 0, are given by the quadratic formula
x=
–b+
b2 – 4ac
2a
(note: this formula is derived via the “completing
the square” method…)
Guided Practice:
2
Solve for x (using the quad formula): 3x – 6x = 5
Solution:
x=1+
2 6
3
= 2.633, –0.633
 Can we support this
answer graphically?
Guided Practice:
Solve by using the quadratic formula:
2 x  3x  1  0
2
Solution:
1
x  1,
2
Other Types of Problems…
Solve the equation graphically and with a table:
3
x –x–1=0
x = 1.325
Not an exact answer  rounded to the thousandth
Other Types of Problems…
Solve the equation graphically(this may be done 2
different ways graphically):
2
5x – 6x – 23 = 0
Remember to show your graph as part of your solution!
Solution: x = –1.627, 2.827
Other Types of Problems…
Solve the equation algebraically (support graphically):
|2x – 1| = 6
Solution: x = 3.5, –2.5
Graphical Support???
Other Types of Problems…
Solve the equation graphically (think intersections
of 2 separate equations):
x
2
= |2x – 3|
Solution: x = –3, 1
Other Types of Problems…
A particular rugby pitch is 30 meters longer than it is wide,
and the area of the pitch is 8800 m 2. What are the dimensions
of this particular pitch?
Area: (length)(width) = 8800
But here, length = width + 30…
 w  30 w  8800
w  30w  8800  0
2
 w  80
Solution: 80 meters x 110 meters
Whiteboard Problems:
Solve by using the quadratic formula (not the program):
x  5  3x
2
Solution:
x  1.532,3.264
Whiteboard Problems:
Solve the equation graphically:
1
|3x – 2| =
x–1
2
No Solution!
Why??
Whiteboard Problems:
2
Solve by graphing: 3x + 2x – 9 = 0
Solution:
x = 1.431, –2.097
 How could we get exact answers???
 Quadratic equation (not the program)!
Whiteboard Problems:
Solve by completing the square:
2
3x – 6x – 7 =0
4
46
Solution: x  
3
3
Whiteboard problems…
Solve the equation graphically:
1
3
3
x =x–
1
2
Solution: x = –1.942, 0.558, 1.384
Homework: p. 50-51 1-23 odd,
31, 43
• Quiz tomorrow on sections 3, 4,
and 5 !!!
• Remember I’m here before school
if you need help on some
material!