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Algebra
Solving Equations Containing x2
Key Questions
What is the inverse operation of “raising to an exponent”?
How do I solve equations that contain x2?
Tasks
As you read this lesson, keep in mind the key questions that are given above. When
you have finished the lesson, you will be asked to complete the following three tasks:
1. Solve three equations involving x2.
2. Make a flow map that can be used to solve equations containing x2.
3. Complete a writing assignment that explains how to solve a given equation
involving x2.
Lesson
At the beginning of Unit Two we learned that we can solve equations for a variable by
using inverse operations to isolate the variable. For example, we looked at the
equation 3x + 7 = 25. We noticed that there are two operations on the left hand side
of the equal sign that prevent the variable x from being isolated. The variable is
being multiplied by 3, and then that answer has 7 added to it. The inverse operation
that “undoes” multiplying by 3 is dividing by 3. The inverse operation that undoes
the addition of 7 is subtraction of 7. In order to isolate x we eventually decided that
these inverse operations needed to be performed in reverse order of the Order of
Operations. So, our solution looked like the following:
3x + 7 = 25
-7 -7
3x = 18
3
3
x =6
In this lesson, we are going to make a slight change to the equation to create the
following new equation:
3x2 + 7 = 25
Now we have one more operation involving the variable x that needs to be undone in
order to isolate x; the operation of “raising to an exponent”. Let’s assume that in
order to isolate x we will still perform our inverse operations in reverse order of the
Order of Operations. This means that the x2 will be the last operation to be undone
since it is higher on the Order of Operations than multiplication. Therefore, our
solution will look as follows:
3x2 + 7 = 25
-7 -7
3x2 = 18
3
3
2
x =6
Algebra
Solving Equations Containing x2
The only thing that we need to do to isolate x is to “undo” the x 2. To see how we do
this, let’s consider a more obvious example:
x2 = 9
Ask yourself the following question: What number multiplied by itself is equal to 9?
This is actually a trick question since there are two such numbers: x = 3 and x = -3.
Remember, (-3)(-3) = 9 AND (3)(3) = 9.
The answer to the equation x2 = 9 is fairly obvious to most people because x = 3 is a
nice, whole number. The solution to x2 = 6 is less obvious because there is no whole
number that can be multiplied by itself to equal 6. This leads us to the need for an
inverse operation to undo the x2. As it turns out, the inverse operation of “raising to
an exponent” is “taking a root”. In this particular case, the inverse operation of x 2 is
x . So, let’s go back to our example:
3x2 + 7 = 25
-7 -7
3x2 = 18
3
3
2
x =6
At this point, to get x by itself, we take the square root of both sides of the equation.
x2  6
On the left hand side of the equation, the square root undoes the x 2, leaving us with x
all by itself. On the right hand side, we find the square root of 6 by using our
calculator as follows:
Remember how the equation x2 = 9 had two solutions? The same thing is true for all
equations where we need to undo x2 by using the square root. In this case, the two
solutions to our equation are x  2.449 and x  -2.449. We often write both of these
solutions together in one expression as follows:
x  2.449
This is read “x is approximately equal to positive or negative 2.449”
Algebra
Solving Equations Containing x2
Let’s look at a few more examples.
Example #1
5x2 - 2 = 18
+2 +2
5x2 = 20
5
5
2
x =4
x2  4
x  2
Example #2
3(4x2 – 5) + 3 = 24
12x2 – 15 + 3 = 24
12x2 – 12 = 24
+12 +12
12x2 = 36
12
12
x2 = 3
x2  3
x   1.732
Example #3
2(4 – 3x2) + 3(5x2 – 6) = 8
8 – 6x2 + 15x2 – 18 = 8
9x2 – 10 = 8
+10 +10
9x2 = 18
9
9
2
x =2
x2  2
x   1.414
Algebra
Solving Equations Containing x2
Task One [3 pts.]
Solve each of the following equations in the space provide below. Make sure that you
show all of your work.
8x2 + 20 = 36
12 – 3(5 – 2x2) = 39
4(3x2 + 5) – 2(4x2 – 7) = 50
Task Two [5 pts.]
In the space below, make a flow map that can be used to solve equations that contain
x2. Use the examples from this lesson to make sure that your flow map is COMPLETE.
Also, make sure that your flow map is GENERAL and EASY TO USE.
.
Task Three (7 pts.]
In writing, describe how to solve an equation involving x2 using the third problem from
Task One as your example.