Question

If alpha and beta are the zeroes of the polynomial \[{x^2} - 3x + k\], such that alpha-beta = 1, find the value of $k$.

Answer 1

Hint: Here in this question given a quadratic equation then \[\alpha \] and \[\beta \]are the zeros of polynomial, we have to find the value of k using the concept of zeros of polynomials. First we need to find the sum of zeroes is equal to \[\dfrac{{ - b}}{a}\] and product of zeroes is equal to \[\dfrac{c}{a}\], where \[a\] is the coefficient of \[{x^2}\], b is the coefficient of \[x\] and \[c\] is the constant and then using the given condition \[\alpha - \beta = 1\] on simplification we get the required value of $k$.

Complete step by step answer:
In General, the quadratic equation in the form of \[a{x^2} + bx + c = 0\]
Where, ‘\[a\]’ is the coefficient of \[{x^2}\], ‘\[b\]’ is the coefficient of \[x\] and ‘\[c\]’ is the constant term.
Similarly, as we know that
Sum of zeroes\[ = \dfrac{{ - b}}{a}\]
Product of zeroes \[ = \dfrac{c}{a}\]
Now Consider the question, given, \[\alpha \] and \[\beta \] are the zeros of the polynomial \[{x^2} - 3x + k\]. We have to find a value of k.
Also given one more condition i.e.,
\[\alpha - \beta = 1\] --- (1)

Consider given polynomial
\[ \Rightarrow \,\,{x^2} - 3x + k\]
Here, the value of \[a = 1\], \[b = - 3\] and \[c = k\]
Given, \[\alpha \] and \[\beta \] are the zeros of the polynomial, then
Let us calculate the sum of the zeros
\[ \Rightarrow \,\,\,\alpha + \beta = \dfrac{{ - \left( { - 3} \right)}}{1}\]
\[\therefore \,\,\alpha + \beta = 3\]---(2)
Let us calculate the product of the zeros
\[ \Rightarrow \,\,\,\alpha \cdot \beta = \dfrac{k}{1}\]
\[\therefore \,\,\,\alpha \cdot \beta = k\] ----(3)

We know that \[{\left( {x + y} \right)^2} - {\left( {x - y} \right)^2} = 4xy\], then
\[ \Rightarrow \,\,{\left( {\alpha + \beta } \right)^2} - {\left( {\alpha - \beta } \right)^2} = 4\,\alpha \cdot \beta \]
Substitute the values from equation (1) , (2) and (3), then we have
\[ \Rightarrow \,\,{\left( 3 \right)^2} - {\left( 1 \right)^2} = 4\,k\]
\[ \Rightarrow \,\,9 - 1 = 4\,k\]
\[ \Rightarrow \,\,8 = 4\,k\]
Divide both side by 4, then we get
\[ \Rightarrow \,\,\dfrac{8}{4} = \,k\]
\[ \Rightarrow \,\,2 = \,k\]
\[\therefore \,\,k = \,2\]

Hence, the value of \[k = \,2\].

Note: The sum and product of the zeros of any quadratic polynomial or equation that play a major role by determining the coefficients and characteristics of any polynomial.Remember, for any polynomial the sum of the zeroes is equal to the negative of the coefficient of \[x\] by the coefficient of \[{x^2}\] and the product of the zeroes is equal to the constant term by the coefficient of \[{x^2}\].