Question

If the product of the zeros of the polynomial \[a{x^2} - 6x - 6\] is 4, find the value of a.

Answer 1

Hint: The polynomial is quadratic. Recall the properties of the quadratic polynomials. The standard form of the quadratic polynomial is \[a{x^2} + bx + c = 0\]. The product of the zeros of the polynomial is equal to \[\dfrac{c}{a}\].

Complete step-by-step answer:
The given equation \[a{x^2} - 6x - 6\] has the highest power or degree as two. Hence, it is a quadratic equation.
A quadratic equation is an equation with the degree or the highest power as two. It contains at least one term with the second power of the variable and it is the highest power in the expression.
The standard form of the quadratic equation is expressed as follows:
\[a{x^2} + bx + c = 0...........(1)\]
zeros of the polynomial are the values of the variable at which the value of the expression becomes zero. It is also called the root or the solution of the polynomial. A polynomial has as many roots equal to its degree. Therefore, the quadratic equation has two roots which can be equal or unequal, real or complex.
We also know that the sum of the roots of a quadratic equation is equal to \[ - \dfrac{b}{a}\] and the product of the roots is equal to \[\dfrac{c}{a}\].
It is given that the product of the roots of the polynomial \[a{x^2} - 6x - 6\] is 4.
From equation (1), we have:
\[a = a;b = - 6;c = - 6..........(2)\]
\[\dfrac{c}{a} = 4\]
From equation (2), we have:
\[\dfrac{{ - 6}}{a} = 4\]
Solving for a, we have:
\[a = \dfrac{{ - 6}}{4}\]
\[a = - \dfrac{3}{2}\]
Hence, the value of a is \[ - \dfrac{3}{2}\].

Note: You can also attempt to find the roots of the polynomial and then multiply them and equate to
4 and then solve to find the value of a. But, this method would be time consuming.