Copyright © 2011 Pearson, Inc. P.5 Solving Equations Graphically , Numerically and Algebraical
Dec 13, 2015
Slide P.5 - 2 Copyright © 2011 Pearson, Inc.
What you’ll learn about
Solving Equations Graphically Solving Quadratic Equations Approximating Solutions of Equations Graphically Approximating Solutions of Equations Numerically
with Tables Solving Equations by Finding Intersections
… and whyThese basic techniques are involved in using a graphing
utility to solve equations in this textbook.
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Example Solving by Finding x-Intercepts
2Solve the equation 2 3 2 0 graphically.x x− − =
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Solution
2Find the -intercepts of 2 3 2.
Use the Trace to see that ( 0.5,0) and (2,0) are -intercepts.
Thus the solutions are 0.5 and 2.
x y x x
x
x x
= − −−
=− =
2Solve the equation 2 3 2 0 graphically.x x− − =
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Zero Factor Property
Let a and b be real numbers.
If ab = 0, then a = 0 or b = 0.
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Quadratic Equation in x
A quadratic equation in x is one that can be written in the form
ax2 + bx + c = 0,
where a, b, and c are real numbers with a ≠ 0.
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Completing the Square
To solve x 2 +bx=c by completing the square, add (b/ 2)2 toboth sides of the equation and factor the left side of the new equation.
x2 +bx+b2
⎛
⎝⎜⎞
⎠⎟
2
=c+b2
⎛
⎝⎜⎞
⎠⎟
2
x+b2
⎛
⎝⎜⎞
⎠⎟
2
=c+b2
4
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Quadratic Formula
The solutions of the quadratic equation ax 2 +bx+c=0, where a≠0, are given by the quadratic formula
x=−b± b2 −4ac
2a.
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Example Solving Using the Quadratic Formula
2Solve the equation 2 3 5 0.x x+ − =
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Solution
a =2, b=3, c=−5
x=−b± b2 −4ac
2a=−3± 32 −4 2( ) −5( )
2 2( )
=−3± 49
4=−3±74
x=−52 or x=1.
2Solve the equation 2 3 5 0.x x+ − =
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Solving Quadratic Equations Algebraically
These are four basic ways to solve quadratic equations algebraically.
1. Factoring
2. Extracting Square Roots
3. Completing the Square
4. Using the Quadratic Formula
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Agreement about Approximate Solutions
For applications, round to a value that is reasonable for the context of the problem. For all others round to two decimal places unless directed otherwise.
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Example Solving by Finding Intersections
Solve the equation −2 x−2 =−3.
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Solution
Graph y =−2 x−2 and y=−3.
Use Trace or the intersectfeature of your grapher tofind the points of intersection.The graph indicates that thesolutions arex=0.5 and x=3.5.
Solve the equation −2 x−2 =−3.
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Quick Review
( )( )( )
2
3 2
4 2
Expand the product.
1. 2
2. 2 1 4 3
Factor completely.
3. 2 2
4. 5 36
5. Combine the fractions and reduce the resulting fraction
2to lowest terms.
2 1 1
x y
x x
x x x
y y
x
x x
+
+ −
+ − −
+ −
−+ −
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Quick Review Solutions
( )( )( )
( )( )( )( )( )( )
2 2
3 2
4 2 2
2
2
4 4
8 2 3
1
Expand the product.
1. 2
2. 2 1 4 3
Factor completely.
3. 2 2
4. 5 36
5. Combine the fractions and reduce the resulting fraction
to low
2
9 2
e
1
2
x xy y
x x
x x x
y y
x y
x x
x x
y y
x
y
+
+ −
+ − −
+
+ +
− −
+ − +
+ − +−
( )( )
22st terms.
2 1 11
5 2
2 1
xx
x x
x
x x−
+ −
− +
+ −