Question

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:
2x² – x – 6

Answer 1

Let f(x) = 2x² – x – 6

$$\begin{gathered} \mathrm{To}\mathrm{find}\mathrm{the}\mathrm{zeroes},\mathrm{we}\mathrm{put}f\left(x\right)=0 \\ \Rightarrow 2x^2-x-6=0 \\ \Rightarrow 2x^2-4x+3x-6=0 \\ \Rightarrow 2x\left(x-2\right)+3\left(x-2\right)=0 \\ \Rightarrow \left(x-2\right)\left(2x+3\right)=0 \\ \Rightarrow \left(x-2\right)=0\mathrm{or}\left(2x+3\right)=0 \\ \Rightarrow x=2,-\frac{3}{2} \end{gathered}$$

Hence, all the zeroes of the polynomial f(x) are $2\mathrm{and}-\frac{3}{2}$.

Now,
$$\begin{gathered} f\left(2\right)=2{\left(2\right)}^2-2-6 \\ =2\left(4\right)-8 \\ =8-8 \\ =0 \\ f\left(-\frac{3}{2}\right)=2{\left(-\frac{3}{2}\right)}^2-\left(-\frac{3}{2}\right)-6 \\ =2\left(\frac{9}{4}\right)+\frac{3}{2}-6 \\ =\frac{9}{2}+\frac{3}{2}-6 \\ =\frac{12}{2}-6 \\ =6-6 \\ =0 \end{gathered}$$

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