Method 1: Using the fact that the exponent on x, 1, is twice the exponent on . This makes the equation in quadratic form for
Next we can factor this (or use the Quadratic Formula):
If you have trouble seeing this, then try using a temporary variable.
Let q =
Then
And the equation becomes
The factoring is easier to see this way:
Replacing the q with we end up with the same equation we had before the temporary variable:
From the Zero Product Property we know that this (or any) product can be zero only if one (or more) of the factors is zero. So: or
Solving these: or
Square both sides of both equations:
9x = 1 or x = 4
Divide the first equation by 9: or x = 4
Method 2: Solve this as a square root equation (ignoring the fact that the equation is of quadratic form.
Step 1: Isolate the square root:
Square both sides:
On the left side we need to use FOIL or the pattern to square 3x+2.
Simplifying we get:
Step 2: Once the square roots are gone, use appropriate techniques to solve the resulting equation. This equation is a quadratic equation so we want one side to be zero. Subtract 49x from each side we get:
Now we factor (or use the Quadratic Formula):
From the Zero Product Property we know that this (or any) product can be zero only if one (or more) of the factors is zero. So:
9x-1 = 0 or x-4 = 0
Dividing the first equation by 9 we get: or x = 4
Since we squared both sides of the equation you must check your answers. This is so because squaring both sides can introduce what are called extraneous solutions. These are solutions which fit the squared equation but do not fit the original equation. (I'll leave the checking for you to do. You should find that both solutions fit the original equation.)
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