Question

If the sum and product of zeroes of the polynomial \[a{x^2} - 5x + c = 0\] are equal to 10 each, find the value of \[a\] and \[c\].

Answer 1

Hint:
Here, we will use the concept of the sum of the roots and the product of roots of a quadratic equation. We will equate the coefficients of the terms in the polynomial with the sum of the roots and solve it further to get the value of \[a\]. Again equation the coefficients of the terms with the product of the roots and substituting the value of \[a\] we will get the value of \[c\].
Formula used: We will use the following formula:
1) Sum of the roots \[\alpha + \beta = - \dfrac{{{\text{Coefficient of }}x}}{{{\text{Coefficient of }}{x^2}}}\]
2) Product of the roots \[\alpha \beta = \dfrac{{{\text{Constant term}}}}{{{\text{Coefficient of }}{x^2}}}\]

Complete step by step solution:
We are given a polynomial \[a{x^2} - 5x + c = 0\].
Let \[\alpha \] and \[\beta \] be the zeroes of the polynomial \[f\left( x \right)\] respectively.
We are given that the Sum of the roots, \[\alpha + \beta = 10\] and the product of the roots, \[\alpha \beta = 10\] .
So, the coefficient of \[{x^2} = a\], coefficient of \[x = - 5\] and Constant term \[ = c\]
By substituting the coefficient of \[x\] and the coefficient of \[{x^2}\] in the formula \[\alpha + \beta = - \dfrac{{{\text{Coefficient of }}x}}{{{\text{Coefficient of }}{x^2}}}\], we get
\[\alpha + \beta = - \dfrac{{ - 5}}{a}\]
We know that Sum of the roots \[\alpha + \beta = 10\], so we get
\[ \Rightarrow \dfrac{5}{a} = 10\]
On cross multiplication, we get
\[ \Rightarrow a = \dfrac{5}{{10}}\]
\[ \Rightarrow a = \dfrac{1}{2}\]
By substituting the constant term and the coefficient of \[{x^2}\] in the formula \[\alpha \beta = \dfrac{{{\text{Constant term}}}}{{{\text{Coefficient of }}{x^2}}}\], we get
 \[\alpha \beta = \dfrac{c}{a}\]
We know that the product of the roots \[\alpha \beta = 10\].
\[ \Rightarrow \dfrac{c}{a} = 10\]
Substituting \[a = \dfrac{1}{2}\] in the above equation, we get
 \[ \Rightarrow \dfrac{c}{{\dfrac{1}{2}}} = 10\]
By rewriting the equation, we get
\[ \Rightarrow 2c = 10\]
Dividing by 2, we get
\[ \Rightarrow c = \dfrac{{10}}{2}\]
\[ \Rightarrow c = 5\]

Therefore, the value of \[a = \dfrac{1}{2}\] and \[c = 5\].

Note:
We are given a polynomial which is of the form of the quadratic equation. A quadratic equation is an equation with the highest degree as 2. If we need to find the product of the roots, then the constant term and the coefficient of \[{x^2}\] has to be multiplied and if we need to find the sum of the products, then it must be equal to the coefficient of \[x\].