Question

If $\alpha {\text{ and }}\beta $ are the roots of ${x^2} - ax + {b^2} = 0$ , then ${\alpha ^2} + {\beta ^2}$ is equal to
\[
  A.{\text{ }}{a^2} + 2{b^2} \\
  B.{\text{ }}{a^2} - 2{b^2} \\
  C.{\text{ }}{a^2} - 2b \\
  D.{\text{ }}{a^2} + 2b \\
  E.{\text{ }}{a^2} - {b^2} \\
 \]

Answer 1

Hint- in order to solve this problem first use the properties of roots of quadratic equation and find some relation between the roots of quadratic equation. Then use the algebraic identities to establish some relation between the problem function and the terms found out from the quadratic equation.

Complete step-by-step answer:
Given that quadratic equation is ${x^2} - ax + {b^2} = 0$
And the roots of this quadratic equation are $\alpha {\text{ and }}\beta $
As we know that for the general quadratic equation of the form $m{x^2} + nx + s = 0$ having some roots p and q. The sum and the product of the roots are given by:
$
  p + q = \dfrac{{ - n}}{m} \\
  p \times q = \dfrac{s}{m} \\
 $
So using the same property for the given quadratic equation let us first compare the general form and use the property.
So we have
$
  m = 1 \\
  n = - a \\
  s = {b^2} \\
  p = \alpha \\
  q = \beta \\
 $
So using the above property we get:
Sum of the roots:
$\alpha + \beta = \dfrac{{ - \left( { - a} \right)}}{1} = a$ ……….. (1)
Product of the roots:
$\alpha \times \beta = \dfrac{{{b^2}}}{1} = {b^2}$ ……………. (2)
Now we have 2 equations. Let us use the algebraic identity connecting equation (1), equation (2) and the problem term.
As we know the algebraic identity
${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$
Let us substitute a and b by the following term $\alpha = a\ & \beta = b$ in the given equation. We get:
${\left( {\alpha + \beta } \right)^2} = {\alpha ^2} + {\beta ^2} + 2\alpha \beta $
Now let us modify the equation bringing the known terms to RHS
$ \Rightarrow {\alpha ^2} + {\beta ^2} = {\left( {\alpha + \beta } \right)^2} - 2\alpha \beta $
Now let us substitute the values from equation (1) and equation (2)
\[
   \Rightarrow {\alpha ^2} + {\beta ^2} = {\left( a \right)^2} - 2\left( {{b^2}} \right) \\
   \Rightarrow {\alpha ^2} + {\beta ^2} = {a^2} - 2{b^2} \\
 \]
Hence, the value of ${\alpha ^2} + {\beta ^2}$ is equal to \[{a^2} - 2{b^2}\]
So, option B is the correct option.

Note- This problem can also be solved by finding the value of both the roots of the quadratic equation that is $\alpha {\text{ and }}\beta $ separately then substituting in the given problem term. But that would be a bit difficult. Students must try to use the algebraic identities to solve such problems. Also the relation between the roots of the quadratic equation are very important and must be remembered.