Question 616469
{{{sqrt(x-3) = 6-x}}}
Since the square root is by itself already we can go ahead and square both sides:
{{{(sqrt(x-3))^2 = (6-x)^2}}}
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 {{{(a-b)^2 = a^2-2ab+b^3}}} pattern. I like using the pattern:
{{{x-3 = (6)^2-2(6)(x) + (x)^2}}}
which simplifies to:
{{{x-3 = x^2-12x + 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^2 -13x + 39}}}<br>
Next we factor or use the Quadratic Formula. This won't factor so we have to use the formula:
{{{x = (-(-13) +- sqrt((-13)^2 -4(1)(39)))/2(1)}}}
Simplifying...
{{{x = (-(-13) +- sqrt(169 -4(1)(39)))/2(1)}}}
{{{x = (-(-13) +- sqrt(169 -156))/2(1)}}}
{{{x = (-(-13) +- sqrt(13))/2(1)}}}
{{{x = (13 +- sqrt(13))/2}}}<br>
So {{{x = (13 + sqrt(13))/2}}} or {{{x = (13 - sqrt(13))/2}}}<br>
Since we squared both sides of the equation, which is not a mistake, we <i>must</i> check out solutions. Use the original equation:
{{{sqrt(x-3) = 6-x}}}
Checking {{{x = (13 + sqrt(13))/2}}}
{{{sqrt(((13 + sqrt(13))/2)-3) = 6-((13 + sqrt(13))/2)}}}
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.)<br>
Checking {{{x = (13 - sqrt(13))/2}}}
{{{sqrt(((13 - sqrt(13))/2)-3) = 6-((13 - sqrt(13))/2)}}}
{{{sqrt(((13 - sqrt(13))/2)-6/2) = 12/2-((13 - sqrt(13))/2)}}}
{{{sqrt((7 - sqrt(13))/2) = (-1 + sqrt(13))/2)}}}
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((7 - sqrt(13))/2) = sqrt(13)/2 - 1/2)}}}
Squaring both sides:
{{{(sqrt((7 - sqrt(13))/2))^2 = (sqrt(13)/2 - 1/2)^2)}}}
{{{(7 - sqrt(13))/2 = (sqrt(13)/2)^2 - 2(sqrt(13)/2)(1/2) + (1/2)^2}}}
{{{7/2 - sqrt(13)/2 = 13/4 - sqrt(13)/2 + 1/4}}}
{{{7/2 - sqrt(13)/2 = 14/4 - sqrt(13)/2}}}
{{{7/2 - sqrt(13)/2 = 7/2 - sqrt(13)/2}}}
Check!<br>
So the only solution to your equation is: {{{x = (13 - sqrt(13))/2}}}<br>
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.