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Question: If we are given that \(\alpha + \beta = 3\) and \({\alpha ^3} + {\beta ^3} = 7\), then show that \(\...

If we are given that α+β=3\alpha + \beta = 3 and α3+β3=7{\alpha ^3} + {\beta ^3} = 7, then show that α\alpha and β\beta are the roots of 9x2−27x+20=09{x^2} - 27x + 20 = 0.

Explanation

Solution

To attempt this question the knowledge of the concept of quadratic equation is must and also remember to use algebraic formula like a3+b3=(a+b)(a2+b2−ab){a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} + {b^2} - ab} \right) and a2+b2+2ab=(a+b)2{a^2} + {b^2} + 2ab = {\left( {a + b} \right)^2} to simplify the equation, using this information you can approach the solution.

Complete step-by-step solution:
Now it is been given that α+β=3\alpha + \beta = 3 and α3+β3=7{\alpha ^3} + {\beta ^3} = 7. We need to show that α\alpha and β\beta are the roots of 9x2−27x+20=09{x^2} - 27x + 20 = 0
Now α3+β3=7{\alpha ^3} + {\beta ^3} = 7 so using a3+b3=(a+b)(a2+b2−ab){a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} + {b^2} - ab} \right)
We can say that α3+β3=(α+β)(α2+β2−αβ){\alpha ^3} + {\beta ^3} = \left( {\alpha + \beta } \right)\left( {{\alpha ^2} + {\beta ^2} - \alpha \beta } \right)
Putting values from above we get
7=3(α2+β2−αβ)7 = 3\left( {{\alpha ^2} + {\beta ^2} - \alpha \beta } \right)
(α2+β2−αβ)=73\left( {{\alpha ^2} + {\beta ^2} - \alpha \beta } \right) = \dfrac{7}{3}
Now (α2+β2−αβ)\left( {{\alpha ^2} + {\beta ^2} - \alpha \beta } \right)
Since we know that a2+b2+2ab=(a+b)2{a^2} + {b^2} + 2ab = {\left( {a + b} \right)^2}
So, we can rewrite the above equation as (α+β)2−3αβ{\left( {\alpha + \beta } \right)^2} - 3\alpha \beta
(α+β)2−3αβ=73{\left( {\alpha + \beta } \right)^2} - 3\alpha \beta = \dfrac{7}{3}
Again, substituting the values, we get
9−73=3αβ9 - \dfrac{7}{3} = 3\alpha \beta
Hence αβ=209\alpha \beta = \dfrac{{20}}{9}
Now if sum of roots and product of roots is given that the quadratic equation can be written as x2−(sum of roots)x + product = 0{x^2} - (sum{\text{ of roots)x + product = 0}}
Thus, the quadratic equation having roots α\alpha and β\beta is x2−(α+β)x+αβ=0{x^2} - (\alpha + \beta )x + \alpha \beta = 0
So, putting values we get
x2−3x+209=0{x^2} - 3x + \dfrac{{20}}{9} = 0
Therefore, on solving we get
9x2−27x+20=09{x^2} - 27x + 20 = 0 which is the desired equation.

Note: Whenever we face such problems the key concept that needs to be in our mind is if somehow, we get the sum and the product of the roots then we can easily get the required quadratic equation. Hence simply accordingly to obtain sum and product of roots.