Question

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
$x^2+7x+12$

Answer 1

$$\begin{gathered} x^2+7x+12=0 \\ \Rightarrow x^2+4x+3x+12=0 \\ \Rightarrow x\left(x+4\right)+3\left(x+4\right)=0 \end{gathered}$$

$$\begin{gathered} \Rightarrow \left(x+4\right)\left(x+3\right)=0 \\ \Rightarrow \left(x+4\right)=0\mathrm{or}\left(x+3\right)=0 \\ \Rightarrow x=-4\mathrm{or}x=-3 \end{gathered}$$

Sum of zeroes = $-4+\left(-3\right)=\frac{-7}{1}=\frac{-\left(\mathrm{coefficient}\mathrm{of}x\right)}{\left(\mathrm{coefficient}\mathrm{of}{x}^2\right)}$
Product of zeroes = $\left(-4\right)\left(-3\right)=\frac{12}{1}=\frac{\mathrm{constant}\mathrm{term}}{\mathrm{coefficient}\mathrm{of}{x}^2}$