quadratic equations c.a.1-3. solving quadratic equations if a quadratic equation ax²+bx+c=0 can’t...
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Quadratic Equations
C.A.1-3
Solving Quadratic Equations
• If a quadratic equation ax²+bx+c=0 can’t be solved by factoring that means the solutions involve roots.
• The quadratic formula can be used to solve any quadratic equation by using the coefficients a, b, and c.
02 cbxaxa
cabbx
2
42
*Since the Quadratic Formula involves the square root we can have a variety of different solutions; from rational to radical to complex.
The derivation of the Quadratic Formula
Using the quadratic formula to solve equations….
• Make sure your equation is in the form, ax²+bx+c=0, if it isn’t then use algebra to put it in this form.
• Identify the coefficients a, b, and c.
• Plug them into the equation and simplify the result.
0352 xx3,5,1 cba
2
135
2
12255
12
31455 2
x
2
135,
2
135
These are the two solutions, they are radical solutions
Examples… 53 xx
053
5553
53
2
2
2
xx
xx
xxSolve:
5,3,1 cba
2
293
2
2093
12
51433 2
x
Examples…Solve: xx 10232
02310
10232
2
xx
xx
252
22
2
10
2
810
2
9210010
12
231410)10( 2
x
23,10,1 cba
*The solutions are approximately 6.4142 and 3.5858 if rounded to 4 decimal places.
Solving Quadratic Inequalities• We can use our factoring and
solution methods to determine when a quadratic has positive and negative output values.
• First find the zeros for the quadratic.
• Place them on a number line and test values on each side of the zeros to determine the sign of the region.
• + means the region is positive• - means the regions is
negative• List all of the regions that
satisfy the inequality in interval notation.
012 xx
includednot
xzeros 1,2:
Try x=-2
+Try x=0
-Try x=3
+-1 2
Solution: (-∞,-1)U(2,∞)
Examples….
Solve the inequality:
01032 xx
025 xx
includednot
xzeros 2,5:
Try x=-3
+Try x=0
-Try x=6
+
-2 5
Solution: (-2,5)
Examples….
Solve the inequality:
0562 xx
051 xx
included
xzeros 1,5:
Try x=-6
+Try x=-2
-Try x=0
+
-5 -1
Solution: (-∞,-5]U[-1,∞)
Inverses-3-7 topic
p.305-313
Finding an inverse for a function…
• For a function f(x)=rule, put in y=rule form.• Swap y with x and vice versa.• Solve for y.• The inverse is the function y=new rule and is
denoted:
rulenewxf 1
Examples….
1. Find the inverse for f(x)=2x+3.
2
32
3
23
32
32
32
1
xxf
yx
yx
yx
xy
xxf
Swap the x’s and y’s!
Solve for y!
Examples….
2. Find the inverse of
53
53
533
53
53
5
1
xxf
yx
yx
yx
xy
xxf
3
5x
xf
Swap the x’s and y’s!
Solve for y!