To find the quadratic equation with roots 5 and 7, we need to use the fact that if a quadratic equation has roots [tex]\(a\)[/tex] and [tex]\(b\)[/tex], it can be expressed as:
[tex]\[ (x - a)(x - b) = 0 \][/tex]
Given that the roots are 5 and 7, we start by writing:
[tex]\[ (x - 5)(x - 7) = 0 \][/tex]
Next, we expand this expression:
[tex]\[ (x - 5)(x - 7) \][/tex]
[tex]\[ = x(x - 7) - 5(x - 7) \][/tex]
[tex]\[ = x^2 - 7x - 5x + 35 \][/tex]
[tex]\[ = x^2 - 12x + 35 \][/tex]
Therefore, the quadratic equation with roots 5 and 7 is:
[tex]\[ y = x^2 - 12x + 35 \][/tex]
Looking at the given options, we see that this matches option J.
So, the correct answer is:
d. J
