SOLUTION: help me solve this please: sqrt(x-3) = (no sqrt anymore) 6-x

Algebra ->  Square-cubic-other-roots -> SOLUTION: help me solve this please: sqrt(x-3) = (no sqrt anymore) 6-x      Log On


   

Question 616469: help me solve this please: sqrt(x-3) = (no sqrt anymore) 6-x
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%28x-3%29+=+6-x
Since the square root is by itself already we can go ahead and square both sides:
%28sqrt%28x-3%29%29%5E2+=+%286-x%29%5E2
Squaring the square root is easy. Squaring the right side can easily be done incorrectly. You must use FOIL on (6-x)(6-x) or use the %28a-b%29%5E2+=+a%5E2-2ab%2Bb%5E3 pattern. I like using the pattern:
x-3+=+%286%29%5E2-2%286%29%28x%29+%2B+%28x%29%5E2
which simplifies to:
x-3+=+x%5E2-12x+%2B+36
Now we have an equation we can solve. It is a quadratic equation so we want one side to be zero. Subtracting x and adding 3 we get:
0+=+x%5E2+-13x+%2B+39

Next we factor or use the Quadratic Formula. This won't factor so we have to use the formula:
x+=+%28-%28-13%29+%2B-+sqrt%28%28-13%29%5E2+-4%281%29%2839%29%29%29%2F2%281%29
Simplifying...
x+=+%28-%28-13%29+%2B-+sqrt%28169+-4%281%29%2839%29%29%29%2F2%281%29
x+=+%28-%28-13%29+%2B-+sqrt%28169+-156%29%29%2F2%281%29
x+=+%28-%28-13%29+%2B-+sqrt%2813%29%29%2F2%281%29
x+=+%2813+%2B-+sqrt%2813%29%29%2F2

So x+=+%2813+%2B+sqrt%2813%29%29%2F2 or x+=+%2813+-+sqrt%2813%29%29%2F2

Since we squared both sides of the equation, which is not a mistake, we must check out solutions. Use the original equation:
sqrt%28x-3%29+=+6-x
Checking x+=+%2813+%2B+sqrt%2813%29%29%2F2
sqrt%28%28%2813+%2B+sqrt%2813%29%29%2F2%29-3%29+=+6-%28%2813+%2B+sqrt%2813%29%29%2F2%29
Looking at the right side...
Since the 6 = 12/2 and since the x value, (13 + sqrt(13))/2, is clearly more than 12/2, the right side turns out negative. The left side is a square root which cannot be negative. So this x value does not check and we reject it. (Note: this answer did not happen because we made a mistake. These "non-solutiosn" can happen any time you square both sides of an equation.)

Checking x+=+%2813+-+sqrt%2813%29%29%2F2
sqrt%28%28%2813+-+sqrt%2813%29%29%2F2%29-3%29+=+6-%28%2813+-+sqrt%2813%29%29%2F2%29
sqrt%28%28%2813+-+sqrt%2813%29%29%2F2%29-6%2F2%29+=+12%2F2-%28%2813+-+sqrt%2813%29%29%2F2%29
sqrt%28%287+-+sqrt%2813%29%29%2F2%29+=+%28-1+%2B+sqrt%2813%29%29%2F2%29
With the square root inside the square root, the rest of the check is bit of a mess. Before we square both sides, let's rewrite the right side as a binomial (so it will be easier to square):
sqrt%28%287+-+sqrt%2813%29%29%2F2%29+=+sqrt%2813%29%2F2+-+1%2F2%29
Squaring both sides:
%28sqrt%28%287+-+sqrt%2813%29%29%2F2%29%29%5E2+=+%28sqrt%2813%29%2F2+-+1%2F2%29%5E2%29

7%2F2+-+sqrt%2813%29%2F2+=+13%2F4+-+sqrt%2813%29%2F2+%2B+1%2F4
7%2F2+-+sqrt%2813%29%2F2+=+14%2F4+-+sqrt%2813%29%2F2
7%2F2+-+sqrt%2813%29%2F2+=+7%2F2+-+sqrt%2813%29%2F2
Check!

So the only solution to your equation is: x+=+%2813+-+sqrt%2813%29%29%2F2

P.S. Sorry about my earlier typo. I typed a -12 for the "b" instead of a -13. When I ended up with complex roots I should have thought to check my typing.