Question

Find the zeroes of the quadratic polynomial \[{{x}^{2}}+7x+12\] and verify the relation between the two zeroes and its coefficients?

Answer 1

Hint: To solve this question, first we will factorize the polynomial \[{{x}^{2}}+7x+12\], by writing 7x as 4x + 3x ad will then evaluate the factors by putting polynomial equal to zero. As the polynomial is quadratic so, we will get 2 roots. Then we will verify the relation between the zeroes and its coefficients which is if we can write \[f(x)={{x}^{2}}-bx+c\] as \[f(x)={{x}^{2}}-(h+k)x+(h\cdot k)\], where h and k are the roots or the zeros of the quadratic equation.

Complete step-by-step answer:
The standard form of a quadratic polynomial is \[f(x)=a{{x}^{2}}+bx+c\] , where a, b, and c are the coefficients of \[{{x}^{2}}\] , x and constants respectively. And the general form of a quadratic polynomial is as follows \[f(x)={{x}^{2}}-bx+c\], where b is the sum of the roots or zeros of the polynomial and c is the product of the roots or zeros of the polynomial. Therefore, the quadratic equation can also be written as follows \[f(x)={{x}^{2}}-(h+k)x+(h\cdot k)\] , where h and k are the roots or the zeros of the quadratic equation.

As mentioned in the question, we have to find the zeros or the roots of the quadratic polynomial. So, we can write the given polynomial as follows
\[\begin{align}
  & ={{x}^{2}}+7x+12 \\
 & =\left( x+4 \right)\left( x+3 \right) \\
\end{align}\]
Hence, the zeros of the given quadratic polynomial are given as
x=-4, -3
Now, to check the relation of the zeros with the coefficients of the quadratic polynomial, we can use the formula of the general form of a quadratic polynomial which is given in the hint as follows
\[f(x)={{x}^{2}}-bx+c\]
So, the given polynomial can be written as
\[f(x)={{x}^{2}}-(-7)x+12\]
Now, from this equation, we can infer that the sum of the roots is -7 and the product of the roots is 12.
Now, the product of roots is
\[(-4)(-3)=12\]
And the sum of the roots is
\[\left( -4 \right)+\left( -3 \right)=-7\]
Hence, the relation has been verified.


Note: The students can make an error if they don’t know the definition of the general form of a quadratic equation which is as follows the general form of a quadratic function is given by
\[f(x)={{x}^{2}}-bx+c\], where b is the sum of the roots or zeros of the polynomial and c is the product of the roots or zeros of the polynomial. Try not to make any calculation mistakes while solving the question.