Question

If \[\alpha \] and \[\beta \] are the roots of the equation \[{{x}^{2}}-6x+a=0\] and satisfy the equation \[3\alpha +2\beta =16\] then the value of \[a\] is?
1. \[-8\]
2. \[8\]
3. \[-16\]
4. \[9\]
5. None of these

Answer 1

Hint: To solve this question one must use the logic and formula of what the product of roots of a quadratic equation and what the sum of roots is. We can use those roots to get two equations for \[\alpha \]and \[\beta \] . Now since we know that there are three variables in this question we will need three different equations to solve this question and be able to find the value of the three variables. Also the formula of product of roots and sum of roots is
Product of roots \[=\dfrac{\text{Constant term}}{\text{Coeff of }{{x}^{2}}}\]
Sum of roots \[=-\dfrac{\text{Coeff of }x}{\text{Coeff of }{{x}^{2}}}\]

Complete step-by-step answer:
Now here in this question we have the equation given to us that
\[{{x}^{2}}-6x+a=0\]
Now here to solve this question we need to know certain things;
Firstly the value of coefficient of \[x\] is found to be \[=1\]
Similarly we can say that in this quadratic equation the coefficient of \[{{x}^{2}}\] is seen to be \[=6\]
Likewise the constant in this quadratic equation is \[=a\]
Now we can use this to find the product and sum of roots. We are given in this question that the two roots are \[\alpha \]and \[\beta \] therefore we can say that
Product of roots: \[\alpha \times \beta =\dfrac{a}{1}\]
Now similarly we can say;
Sum of roots: \[\alpha +\beta =-\dfrac{-6}{1}\]
\[\alpha +\beta =6\]
Now it is given that \[3\alpha +2\beta =16\] therefore we need to simplify and solve the two equations of roots
We have
\[3\alpha +2\beta =16\]
\[3\alpha +3\beta =18\] we get this by \[(\alpha +\beta =6)\times 3\]
Now subtracting both equations the first terms we can see will be cancelled and we are left with
\[-\beta =-2\]
Multiplying by negative one on both sides we get the value
\[\beta =2\]
Now applying value of \[\beta \] in \[\alpha +\beta =6\] which gives us
\[\alpha +2=6\]
Further simplifying we get,
\[\alpha =6-2\]
\[\alpha =4\]
Now to find \[a\]we have the equation that
\[\alpha \times \beta =a\]
Substituting we get
\[a=2\times 4\]
\[a=8\]
So, the correct answer is “Option 2”.

Note: Here a common mistake made is people use the formula \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] where x stands for roots of a quadratic equations and then assume the roots to be either but we can’t do that because it may not be same as the value of the roots they’ve defined in the equation \[3\alpha +2\beta =16\] and we could get the wrong answer.