L1.1 C1 Solving Radical Equations Graphically - Part 1
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Unit 1
Functions
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Part 1.1C corresponds to section 2.3, Solving Radical Equations Graphically, starting on page 90 of your Pre-Calculus 12 textbook.
To solve means to determine a solution or answer to a problem.
To solve an equation means to determine the value(s) of the variable that satisfy the equation. This is commonly referred to as ‘finding the roots of the equation’.
Equations can be solved both graphically and algebraically. Sometimes it is helpful to confirm a solution obtained using one approach by using another approach.
Several strategies can be used when solving equations, either with a graph or with algebra. The following examples illustrate some of those strategies.
To solve an equation means to determine the value(s) of the variable that satisfy the equation. This is commonly referred to as ‘finding the roots of the equation’.
Equations can be solved both graphically and algebraically. Sometimes it is helpful to confirm a solution obtained using one approach by using another approach.
Several strategies can be used when solving equations, either with a graph or with algebra. The following examples illustrate some of those strategies.
Using each of the methods listed below, determine the roots of the equation «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»0«/mn»«mo»=«/mo»«msqrt»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/msqrt»«mo»§#8722;«/mo»«mn»4«/mn»«/mrow»«/mstyle»«/math».
a.
Algebraically
b.
Graphically
a.
Identify the restrictions on the variable, and then solve for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math».
Add «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math» to both sides.
Square both sides.
Subtract «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» from both sides.
The value of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»14«/mn»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math» is not a restricted variable value. As such, the root of the equation is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»14«/mn»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mtd»«mtd»«mo»§#8805;«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«/mtd»«mtd»«mo»§#8805;«/mo»«/mtd»«mtd»«mo»-«/mo»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»0«/mn»«mo»=«/mo»«msqrt»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/msqrt»«mo»§#8722;«/mo»«mn»4«/mn»«/mrow»«/mstyle»«/math»
Add «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math» to both sides.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»4«/mn»«mo»=«/mo»«msqrt»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/msqrt»«/mrow»«/mstyle»«/math»
Square both sides.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»16«/mn»«mo»=«/mo»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math»
Subtract «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» from both sides.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»14«/mn»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math»
The value of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»14«/mn»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math» is not a restricted variable value. As such, the root of the equation is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»14«/mn»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math».
b.
Use a table of values or technology to graph «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«msqrt»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/msqrt»«mo»§#8722;«/mo»«mn»4«/mn»«/mrow»«/mstyle»«/math».
The root of the equation «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»0«/mn»«mo»=«/mo»«msqrt»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/msqrt»«mo»§#8722;«/mo»«mn»4«/mn»«/mrow»«/mstyle»«/math» corresponds to the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercept of the graph of the related function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«msqrt»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/msqrt»«mo»§#8722;«/mo»«mn»4«/mn»«/mrow»«/mstyle»«/math».
Note: If the exact value of the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercept is not clear from the graph, the “Zero” feature of a graphing calculator may be used.
The root of the equation «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»0«/mn»«mo»=«/mo»«msqrt»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/msqrt»«mo»§#8722;«/mo»«mn»4«/mn»«/mrow»«/mstyle»«/math» corresponds to the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercept of the graph of the related function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«msqrt»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/msqrt»«mo»§#8722;«/mo»«mn»4«/mn»«/mrow»«/mstyle»«/math».
Note: If the exact value of the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-intercept is not clear from the graph, the “Zero” feature of a graphing calculator may be used.