Question

If we have two zeros as $\alpha $ and $\beta $ of the polynomial ${{x}^{2}}-5x+m$ such $\alpha -\beta =1$ , then find the value of m.

Answer 1

Hint: Every equation can be written in terms of its zero in the following form: ${{x}^{2}}-(\alpha +\beta )x+\alpha \beta =0$........(i) where $\alpha $ and $\beta $ are zeros of the polynomial. We can compare this with the given equation to find $\alpha $ and $\beta $ and then proceed to find m.

Complete step-by-step solution -
We are given the following equation: ${{x}^{2}}-5x+m=0$ .
We are given the following equation: ${{x}^{2}}-5x+m=0$ . Comparing this equation with equation (i) we have $-(\alpha +\beta )=-5$ and $\alpha \beta =m$ …(ii)
We are given the following equation: $\alpha -\beta =1$ …(iii)
$-(\alpha +\beta )=-5\Rightarrow \alpha +\beta =5$ …(iv)
Adding equation (iiI) and (iv) we have,
$2\alpha =6\Rightarrow \alpha =3$
Now we can use either of equation (ii) and equation (iii) to calculate $\beta $ . From equation (ii) we have, $\beta =\alpha -1$ . Substituting the value of $\alpha $ we have,
$\beta =3-1=2$
Now that we know the value of $\alpha $ and $\beta $ we can calculate the value of m from the equation (ii).
$m=\alpha \beta $
Substituting the value of $\alpha $ and $\beta $ we have,
$m=3\times 2=6$
Hence, the value of m is 6.

Note: The equation ${{x}^{2}}-(\alpha +\beta )x+\alpha \beta =0$ is written when we know the zeroes of the polynomial. In reverse we can calculate the value of $\alpha +\beta $ and $\alpha \beta $ when an equation is known to us.
Suppose the equation $a{{x}^{2}}+bx+c=0$ is given to us and we wish to calculate $\alpha +\beta $ and $\alpha \beta $ then the following formula is used as $\alpha +\beta =\dfrac{-b}{a}$ and $\alpha \beta =\dfrac{c}{a}$ . Both things are basically the same, this formula has been derived in general to directly calculate the sum and product of zeroes of the polynomial.