Question

2. Your friend wants to write the equation of a quadratic function 1 5. 3.r = 1) that has zeros at 3 and 3. They want to use the binomials (3.7" – 5) and You insist that they must use the binomials 3 and (3) Who is correct? Explain.

Answer 1

Let's expand both options, your friend's and yours, as shown below

`\begin{gathered} y=(3x+1)(3x-5)=9x^2-12x-5=9(x^2-(4)/(3)x-(5)/(9)) \\ \text{and} \\ y=(x+(1)/(3))(x-(5)/(3))=x^2-(4)/(3)-(5)/(9) \end{gathered}`

Then, both equations are the same besides a constant that will not affect the zeros of the functions, as shown below

`\begin{gathered} y=(3x+1)(3x-5) \\ x=-(1)/(3) \\ \Rightarrow y=(-1+1)(-1-5)=0 \\ \text{and} \\ x=(5)/(3) \\ \Rightarrow y=(5+1)(5-5)=0 \\ \end{gathered}`

And

`\begin{gathered} y=(x-(5)/(3))(x+(1)/(3)) \\ x=-(1)/(3) \\ \Rightarrow y=(-(1)/(3)-(5)/(3))\cdot0=0 \\ x=(5)/(3) \\ \Rightarrow y=0\cdot((6)/(3))=0 \end{gathered}`

Both your friend and you are correct. The functions are the same with exception of a constant that multiplies the whole function (a scale factor); despite that, the zeros are the same for both functions

`(3x-5)(3x+1)=9(x+(1)/(3))(x-(5)/(3))`