QUESTION

# Find the product of zeroes of ${u}^{2}+8u$.

Hint: We will use the concept of the zeroes of the quadratic equation to find the zeros or roots of the equation. Zeros or roots of a Quadratic equation are the values on which the equation becomes zero.

Complete step-by-step solution -
We know that the general equation of a quadratic polynomial is ax2+bx+c.
To find the zeros of a quadratic equation we equate is equal to zero and find the solution of the resultant equation that is $a{{x}^{2}}+bx+c=0$.
We have to find the zeros of ${{u}^{2}}+8u$. So we will equate this to zero and find the solution of the resultant equation.
${{u}^{2}}+8u$=0 …... (1)
To solve this equation, we will use the method of factorization. In this method, we express the quadratic expression as a multiplication of two factors.
Now, we will take u common in the (1)
u(u+8) =0 …. (2)
Here we have two factors of the quadratic expression as u and u+8.
Now, we know that the product of two factors can be zero only if either of them is zero.
So, we will equate both the factors to zero. This gives:
u=0 or
u+8=0
u=-8
Therefore, we get u=0, -8 as two zeroes of the given quadratic expression.
Therefore, the product of the zeroes is 0*8=0.

Note: Alternative way
We know that a general equation of a quadratic polynomial is $a{{x}^{2}}+bx+c$. The product of zeroes for the general quadratic equation is defined as $c/a$. Where c is constant term and a is the coefficient of ${x}^{2}$
The expression given to us is ${{u}^{2}}+8u$ . This is a quadratic expression in variable u. On comparing it with $a{{x}^{2}}+bx+c$
We get,
a=1
b=8
c=0
Now we know that the product of zeroes is c/a.
Since c=0 we get the product of zeroes as 0.