Question

how to find a and b values

Answer 1

According to given sum,

${a}^{3} - {b}^{3} = (a - b)( {a}^{2} + ab + {b}^{2} )$
$56 = 2( {a}^{2} + ab + {b}^{2} )$
${a}^{2} + ab + {b}^{2} = 28$
given that,

a-b=2

squaring on both sides .

${(a - b)}^{2} = {2}^{2} = 4$
${a}^{2} + {b}^{2} - 2ab = 4$
Now we get two equations. so subtract both the equations. we get

${a}^{2} + ab + {b}^{2} - {a }^{2} - {b}^{2} + 2ab = 28 - 4 = 24$
3ab=24

ab=24/3=8

we have to use the formula.

${(a + b)}^{2} = {(a - b)}^{2} + 4ab$
${(a + b)}^{2} = {2}^{2} + 4(8) = 36$
$a + b = \sqrt{36} = 6$

again we get two equations. so add both equations. we get

a+b+a-b=6+2=8

2a=8

a=4

substitute the value of a in the above equation.

a+b=6

4+b=6

b=2

so
${a}^{2} + {b}^{2} = {6}^{2} + {4}^{2} = 36 + 16$
${a}^{2} + {b}^{2} = 52$

:-)hope it helps u.