Question 1187483
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Solving by trial and error is far easier than a formal algebraic solution.  Simply look for a way to break 40 into the sum of two perfect squares.<br>
1+39?  no
4+36?  YES<br>
The squares of their ages are 4 and 36; their ages are 2 and 6.  6 is 4 more than 2, so that solution works.<br>
But probably a formal algebraic solution was wanted....<br>
Let x = Marie's age
Then x+4 = Anne's age<br>
The sum of the squares of their ages is 40:<br>
{{{x^2+(x+4)^2=40}}}
{{{x^2+x^2+8x+16=40}}}
{{{2x^2+8x-24=0}}}
{{{x^2+4x-12=0}}}
{{{(x+6)(x-2)=0}}}
{{{x=-6}}}  OR  {{{x=2}}}<br>
Obviously the negative solution for Marie's age makes no sense; so x=2.<br>
ANSWERS: Marie's age = x = 2; Anne's age = x+4 = 6<br>