Question

What is the value of $ \left( {\alpha - \beta } \right) $ ?

Answer 1

Hint: : In order to determine the value to the above equation having variables denoted as $ \alpha $ alpha and $ \beta $ beta . First we compare it with the standard quadratic equation i.e. $ a{x^2} + bx + c $ to obtain the value of the $ \left( {\alpha - \beta } \right) $ . We need to know the concept of the Nature of the roots that is Sum of the roots and secondly , the product of the roots . The sum of the roots of a quadratic equation is equal to the negation of the coefficient of the second term, divided by the leading coefficient $ {r_1} + {r_2} = - \dfrac{b}{a} $ . And The product of the roots of a quadratic equation is equal to the constant term (the third term) , divided by the leading coefficient $ {r_1} \times {r_2} = \dfrac{c}{a} $ . By getting an idea of this concept we can easily solve our question given the required answer .

Complete step by step solution:
We need to find the value of $ \left( {\alpha - \beta } \right) $ given .
We are going to denote $ \alpha $ as A and $ \beta $ as B so will now find $ A - B $ .
Let us consider a standard quadratic equation i.e. $ a{x^2} + bx + c $ .
Now as per the concept of sum of the roots , we find the sum of the roots $ A + B $ , for this equation we would get it as $ A + B = - \dfrac{b}{a} $ and product of the roots $ A \times B = \dfrac{c}{a} $ .
Also , we know that –
 $
  {(A - B)^2} = {(A + B)^2} - 4AB \\
   \Rightarrow {(A - B)^2} = {\left( { - \dfrac{b}{a}} \right)^2} - 4\left( {\dfrac{c}{a}} \right) \;
  $
Now if we take LCM here , then we get –
 $ \Rightarrow {(A - B)^2} = \dfrac{{\left( {{b^2} - 4ac} \right)}}{{{a^2}}} $
By taking square root on both the sides of the equation we get –
 $ $ $ \Rightarrow A - B = \dfrac{{\sqrt {\left[ {\left( {{b^2} - 4ac} \right)} \right]} }}{a} $
Hence , the value of $ A - B $ which was we denoted that by $ \left( {\alpha - \beta } \right) $ we get –
 $ \Rightarrow \alpha - \beta = \dfrac{{\sqrt {\left( {{b^2} - 4ac} \right)} }}{a} $
This is our required result .
So, the correct answer is “ $ \dfrac{{\sqrt {\left( {{b^2} - 4ac} \right)} }}{a} $ ”.

Note: One must be careful while calculating the answer as calculation error may come.
Don’t forget to compare the given quadratic equation with the standard one every time.
 Always try to understand the mathematical statement carefully and keep things distinct .
Cross check the answer and always keep the final answer simplified .
Remember the algebraic identities and apply appropriately .