Question

if a plus b equal to 7 , ab equal to 6 find a cube minus b cube

Answer 1

       $\underline{\bold{Answer:}}$

       $+215,-215$

       $\underline{\bold{Step-by-step-explaination:}}$

       $\text{It is given that: }$

       $a+b = 7 \text{ ------(i) }$

       $ab = 6\text{ ------(ii)}$

       $\text{We have to find }a^{3} - b^{3}$

       $\text{We know that, }(x+y)^{2} = x^{2} + y^{2} + 2xy$

$\implies (a+b)^{2} = a^{2} + b^{2} + 2ab$

       $\text{From (i) and (ii)}$

$\implies (7)^{2} = a^{2} + b^{2} + 2(6)$

       $\text{Simplifying...}$

$\implies 49 = a^{2} + b^{2} + 12$

       $\text{Transposing 12 to R.H.S}$

$\implies 49 - 12 = a^{2} + b^{2}$

        $\text{Simplifying...}$

$\implies 37 = a^{2} + b^{2}\text{ ------(iii)}$

       $\text{We know that, }(x-y)^{2} = x^{2} + y^{2} - 2xy$

$\implies (a-b)^{2} = a^{2} + b^{2} - 2ab$

       $\text{From (ii) and (iii)}$

$\implies (a-b)^{2} = 37 - 2(6)$

       $\text{Simplifying...}$

$\implies (a-b)^{2} = 25$

       $\text{Rooting both sides}$

$\implies a-b = \sqrt{25}$

       $\text{Simplifying...}$

$\implies a-b = +5, -5 \text{ ------(iv)}$

       $\text{We know that }x^{3} - y^{3} = (x-y)(x^{2} +y^{2}+xy)$

$\implies a^{3} - b^{3} = (a-b)(a^{2} +b^{2}+ab)$

       $\text{From (ii), (iii) and (iv)}$

$\implies a^{3} - b^{3} = (+5,-5)(37+6)$

       $\text{Simplifying...}$

$\implies a^{3} - b^{3} = (+5,-5)(43)$

       $\text{Simplifying...}$

$\implies \boxed{a^{3} - b^{3} = +215,-215}$