Explain how different quadratic functions can have the same zeros.

Explain how different quadratic functions can have the same zeros.

$\begin{array}{lcl} \text{Quadratic functions with same zeroes are: } a(x-x_1)(x-x_2) = 0 \\ \quad \text{ or } \quad ax^2 + [ -a\cdot (x_1+x_2)] x + a\cdot x_1\cdot x_2 = 0\\ \text{the zeroes are } x_1 \text{ and } x_2 \text{ and } a\ne 0\\\\ \hline \\ \text{Example 1: }\quad a=1\qquad y = x^2+bx+c = 0 \\ b=-(x_1+x_2) \qquad c=x_1\cdot x_2 \qquad \text{the zeroes are } x_1 \text{ and } x_2\\\\ \text{Set } x = x_1: \\ y =(x_1)^2 + [-(x_1+x_2)]\cdot x_1 + x_1\cdot x_2 \\ y =(x_1)^2 - (x_1)^2 - x_2\cdot x_1 + x_1\cdot x_2 = 0\\\\ \text{Set } x = x_2: \\ y =(x_2)^2 + [-(x_1+x_2)]\cdot x_2 + x_1\cdot x_2 \\ y =(x_2)^2 - x_1\cdot x_2 - (x_2)^2 + x_1\cdot x_2 = 0 \\ \hline \\ \text{Example 2: }\quad a=2\qquad y = 2x^2+bx+c = 0 \\ b=-2\cdot (x_1+x_2) \qquad c=2\cdot x_1\cdot x_2 \qquad \text{the zeroes are } x_1 \text{ and } x_2\\\\ \text{Set } x = x_1: \\ y =2(x_1)^2 + [-2(x_1+x_2)]\cdot x_1 + 2x_1\cdot x_2 \\ y =2(x_1)^2 - 2(x_1)^2 - 2x_2\cdot x_1 + 2x_1\cdot x_2 = 0\\\\ \text{Set } x = x_2: \\ y =2(x_2)^2 + [-2(x_1+x_2)]\cdot x_2 + 2x_1\cdot x_2 \\ y =2(x_2)^2 - 2x_1\cdot x_2 - 2(x_2)^2 + 2x_1\cdot x_2 = 0 \end{array}$

$\begin{array}{lcl} \text{Quadratic functions with same zeroes are: }\\\\ \end{array}\\ \begin{array}{rcrcl} && 2x^2+2.2x+0.2&=&0 \qquad \text{the zeroes are } -1 \text{ and } -0.1 \\ 2 &\cdot & (2x^2+2.2x+0.2)=4x^2+4.4x+0.4&=&0 \qquad \text{the zeroes are } -1 \text{ and } -0.1 \\ -2&\cdot & (2x^2+2.2x+0.2)=-4x^2-4.4x-0.4&=&0 \qquad \text{the zeroes are } -1 \text{ and } -0.1 \\ 1.5&\cdot & (2x^2+2.2x+0.2)=3x^2+3.3x+0.3&=&0 \qquad \text{the zeroes are } -1 \text{ and } -0.1 \\ \cdots \end{array}$