Question

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:
3x² – x – 4

Answer 1

Let f(x) = 3x² – x – 4

$$\begin{gathered} \mathrm{To}\mathrm{find}\mathrm{the}\mathrm{zeroes},\mathrm{we}\mathrm{put}f\left(x\right)=0 \\ \Rightarrow 3x^2-x-4=0 \\ \Rightarrow 3x^2-4x+3x-4=0 \\ \Rightarrow x\left(3x-4\right)+1\left(3x-4\right)=0 \\ \Rightarrow \left(x+1\right)\left(3x-4\right)=0 \\ \Rightarrow \left(x+1\right)=0\mathrm{or}\left(3x-4\right)=0 \\ \Rightarrow x=-1,\frac{4}{3} \end{gathered}$$

Hence, all the zeroes of the polynomial f(x) are $-1\mathrm{and}\frac{4}{3}$.

Now,
$$\begin{gathered} f\left(-1\right)=3{\left(-1\right)}^2-\left(-1\right)-4 \\ =3\left(1\right)+1-4 \\ =3-3 \\ =0 \\ f\left(\frac{4}{3}\right)=3{\left(\frac{4}{3}\right)}^2-\frac{4}{3}-4 \\ =3\left(\frac{16}{9}\right)-\frac{4}{3}-4 \\ =\frac{16}{3}-\frac{4}{3}-4 \\ =\frac{12}{3}-4 \\ =4-4 \\ =0 \end{gathered}$$

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