Question

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:
5x² + 10x

Answer 1

Let f(x) = 5x² + 10x

$$\begin{gathered} \mathrm{To}\mathrm{find}\mathrm{the}\mathrm{zeroes},\mathrm{we}\mathrm{put}f\left(x\right)=0 \\ \Rightarrow 5x^2+10x=0 \\ \Rightarrow 5x\left(x+2\right)=0 \\ \Rightarrow \left(x\right)\left(x+2\right)=0 \\ \Rightarrow \left(x\right)=0\mathrm{or}\left(x+2\right)=0 \\ \Rightarrow x=0,-2 \end{gathered}$$

Hence, all the zeroes of the polynomial f(x) are $0\mathrm{and}-2$.

Now,
$$\begin{gathered} f\left(0\right)=5{\left(0\right)}^2+10\left(0\right) \\ =0+0 \\ =0 \\ f\left(-2\right)=5{\left(-2\right)}^2+10\left(-2\right) \\ =5\left(4\right)-20 \\ =20-20 \\ =0 \end{gathered}$$

Hence, the relationship between the zeros and the coefficients is verified.