```Question 470569
Given to solve:
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{{{x + sqrt(x) = 42}}}
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Let's start by isolating the term {{{sqrt(x)}}} on one side of the equation.  That way we can easily square both sides to get rid of the radical sign.
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Begin by subtracting x from both sides of the equation to get:
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{{{sqrt(x) = -x + 42}}}
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Now square both sides of the equation. Note that when you square the radical on the left side, the result is just x and you have:
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{{{x = (-x + 42)^2}}}
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Squaring the right side by multiplying {{{(-x + 42)*(-x + 42)}}} results in the equation becoming:
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{{{x = (-x)^2 -42x -42x + 42^2}}}
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and this further simplifies to:
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{{{x = x^2 - 84x + 1764}}}
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Get this into the standard quadratic form by subtracting {{{x^2 - 84x +1764}}} from both sides and we then have:
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{{{-x^2 + 84x + x -1764 = 0}}}
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and then multiply both sides (all terms) by -1 to change the sign on the squared term to positive:
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{{{x^2 - 84x - x + 1764 = 0}}}
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By combining the two terms containing x we finally get our equation into the standard quadratic form as shown:
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{{{x^2 - 85x + 1764 = 0}}}
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Since this is in the standard form, we can apply the quadratic formula to solve for x.  This formula says that for an equation of the form:
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{{{ax^2 + bx + c = 0}}}
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the two possible values of x are given by:
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{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
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By comparing the terms of the standard form of the quadratic equation with their counterpart terms of the equation that we developed for this problem we can see that the values of a, b, and c for this problem are:
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{{{a = 1}}}
{{{b = -85}}} and
{{{c = 1764}}}
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Substituting these values into the standard form for computing x we get:
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{{{x = (-(-85) +- sqrt( (-85)^2-4*1*1764 ))/(2*1) }}}
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The first term on the right side {{{-(-85)}}} becomes +85 so the equation is then:
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{{{x = (85 +- sqrt( (-85)^2-4*1*1764 ))/(2*1) }}}
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The denominator {{{2*1}}} multiplies out to 2 so the equation for x becomes:
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{{{x = (85 +- sqrt( (-85)^2-4*1*1764 ))/2 }}}
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Under the radical sign the term {{{(-85)^2}}} squares to be 7225 and the term {{{4*1*1764}}} multiplies out to give {{{7056}}}
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Substituting these values into the equation for x gives us:
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{{{x = (85 +- sqrt( 7225-7056 ))/2 }}}
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Subtract the two numbers under the radical to get:
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{{{x = (85 +- sqrt( 169 ))/2 }}}
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The square root of 169 is 13 so the equation is reduced to:
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{{{x = (85 +- 13)/2 }}}
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So we have two possible answers to this problem. First, with the plus sign we get:
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{{{x = 98/2}}} and this results in {{{x = 49}}}
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Then with the minus sign we get:
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{{{x = 72/2}}} and this results in {{{x = 36}}}
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{{{x + sqrt(x) = 42}}}
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and we let {{{x = 49}}} to see if that works. When we substitute 49 for x we get:
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{{{x + sqrt(x) = 42}}}
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{{{49 + sqrt(49) = 42}}}
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The square root of 49 is 7 and substituting this gives us:
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{{{49 + 7 = 42}}}
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Ooops! This does not work because the left side is 56 and the right side is 42 so we need to ignore this answer.
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Next we let {{{x = 36}}} which was our second possible answer. When we substitute 36 for x we get:
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{{{36 + sqrt(36) = 42}}}
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The square root of 36 is 6 so this becomes:
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{{{36 + 6 = 42}}}
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That works!!! So the answer to the original problem is x = 36.
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Hope this helps you to understand the problem and a procedure for solving it. Check all my work to ensure that I didn't accidentally make a dumb mistake.```