You can put this solution on YOUR website!
If this is correct then I do not see an easy solution. (There may be one but I don't see it.) So fasten your seatbelt.
I'll solve this two ways. First I will solve it in a straightforward, conventional way. Then I'll use another method which requires a bit of cleverness.
First method. We start by isolating the square root with the variable in the radicand. Subtracting from each side we get:
Next we square both sides.
Squaring the right side is simple. On the left side we need to use FOIL or the pattern. (I prefer the pattern.):
which simplifies to:
This is a quadratic equation so we want one side to be zero:
Factoring out x from the middle two terms we get:
We can now use the quadratic formula with "a" being 1, "b" being and "c" being 40:
which simplifies as follows:
In long form this is: or
And last of all we need to check these answers. Whenever you have squared both sides of an equation you must check your answers. The two answers above are solutions to the squared equation. But these answers do not necessarily fit the original equation. So even if we have not made any mistakes we may still have one (or more) solutions that do not fit the original equation. So we must check.
To check you must use the original equation. I would suggest using a calculator to find decimal approximations for the two answers above. Then substitute each one into the original equation to see if it fits.
I will leave the check for you. But you should find that the second solution does not actually work. This "solution" is therefore only a solution to the squared equation, not to the original equation. So we reject the second solution above.
Second method. A different way to solve this takes advantage of the fact that there are two terms with "x" and the exponent on "x" (which is 1) is twice as large as the exponent on (which is 1/2 (Remember the fractional exponents?)). Because of theses facts this equation is in "quadratic form for .
As an equation in quadratic form we want one side to be zero. Subtracting the entire right side from both sides we get:
Rewriting the middle term so that it is in terms of we get:
We can now use the Quadratic Formula. However, instead of the formula telling us what x is, it will tell us what is. (If this part confuses you then look at the conclusion to see why this works like this.)
which simplifies as follows:
In long form this is: or
The right side of the second equation is negative. But cannot be negative! So there are no solutions to the second equation. So will just finish solving the first equation.
We want to square both sides. But to make that easier I am going to change things around a bit. Dividing both sides by we get:
which simplifies as follows:
Multiplying both sides by 2 we get:
Since we squared both sides we must once again check. (This time there is only one solution to check. And this is the same solution we found using the other method.)
If you had trouble seeing how the Quadratic Formula can be used on equation like:
then a temporary variable can make it clearer:
Let q =
Then = x. Substituting these into the equation we get:
It should be clear that this is a quadratic equation. Using the Quadratic Formula on this we get:
Substituting back in for q we get:
Once you've done a few of these "quadratic form" equations, it will get easier to see how to solve them.
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