For each location of a zero, call it k, you will have a factor of the quadratic in the form (x - k). Your quadratic, with zeros at x=2 and x=5, would need a factor of (x-2) and (x-5). Putting those together we get the following quadratic: y=a(x-2)(x-5). The variable a in front needs to be there because there are an infinite number of parabolas going through those two zeros. Just picture a very tall slender parabola versus a flatter one. Try typing the formula into desmos.com graphing calculator and add a slider for the variable a. You can see what different values of a do to the parabola.
To find the value of a, plug in x=0 and y=7, which is the location of the given y-intercept.
This gives you 7=a(-2)(-5), which can be simplified to 7=10a. Solving for a gives you a=7/10.
Plug the value of a into the quadratic and your formula would be y = 7/10(x-2)(x-5).
Make sure to verify the answer by graphing or substitution. Hope this helps you understand the process.