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Exponential Equations: Example 9
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Solve
.
Information for
Solving Exponential Equations
How do we proceed?
We need to isolate the exponential.
How can we do that?
We can subtract
from both sides of the equation.
Do it.
What is the next step?
We square to “undo” the exponential.
Do we need to square both sides of the equation?
Yes, we always need to do the same operation on each side of an equation.
Set that up.
Do the multiplication.
We get
Collect like terms.
We still have an exponential. What can we do?
We can square both sides again.
Do that.
We get
Factor this expression.
We get
State the solutions.
How can we check these as possible solutions of the original problem?
We can substitute them into the
original equation
.
Check
x
= 0.
With
x
= 0, we have
What can we conclude?
x
= 0 is a solution of the original equation.
Check
x
= 9.
What can we conclude?
x
= 9 is a solution of the original problem.
Let’s understand this graphically. We’ll rewrite the problem as
If we plot each side, we can see where they cross to compare with our solution.
Plot the left-hand side.
alt="Java applet graph of a hyperbola, foci, vertices, asymptotes." Your browser is completely ignoring the <APPLET> tag!
Add the graph of the right-hand side of the equation.
alt="Java applet graph of a hyperbola, foci, vertices, asymptotes." Your browser is completely ignoring the <APPLET> tag!
Label the points of intersection.
alt="Java applet graph of a hyperbola, foci, vertices, asymptotes." Your browser is completely ignoring the <APPLET> tag!
What is our conclusion?
We conclude that graphing confirms that both
x
= 0 and
x
= 9
are solutions.
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General Contents
Detailed Contents
Index