Question

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients.
$4{{u}^{2}}+8u$

Answer 1

Hint: In this question, first work out on the zeros of the quadratic polynomial. $\left( f(u)=0 \right)$ Second work out on the verification of relation between the zeros and coefficients of the quadratic polynomial. (sum of zeros $=\dfrac{-b}{a}$ and product of zeros $=\dfrac{c}{a}$ ).

Complete step-by-step answer:
Let us consider $f(u)=4{{u}^{2}}+8u$
To find the zeros of the given quadratic polynomial, put $f(u)=0$
$4{{u}^{2}}+8u=0$
Taking the 4u common on left side, we get
$4u\left( u+2 \right)=0$
It gives
$4u=0\text{ or }\left( u+2 \right)=0$
$u=0\text{ or u}=-2$
Hence, the zeros of the given quadratic polynomial are 0 and -2.
Second work out on the verification of relation between the zeros and coefficients of the quadratic polynomial.
Sum of zeros $=\dfrac{-b}{a}$ and product of zeros $=\dfrac{c}{a}$
In the given quadratic polynomial a = 4, b = 8 and c =0
Sum of zeros $=\dfrac{-b}{a}$
$0+(-2)=\dfrac{-8}{4}$
-2 = -2
LHS =RHS.
Also,
Product of zeros $=\dfrac{c}{a}$
$0\times (-2)=\dfrac{0}{4}$
0 = 0
LHS = RHS.
Hence, verified the relation between the zeros and coefficients of the quadratic polynomial.

Note: We might get confused about the difference between the zeros and roots in mathematics. It is the value of x that makes the polynomial equal to 0. In other words, the number r is a root of a polynomial P(x) if and only if P(r) = 0. The word zero refers to the function (polynomial) and the word root refers to the equation.