Question 238424
The key to this problem is to understand that is some number, call it "r", is a root then (x-r) is a factor. So we are looking for an equation of the form:
{{{(x - r[1])(x - r[2]) - 0}}}
that simplifies to
{{{x^2 -12x + p = 0}}}
and where {{{r[1] = r[2] + 2}}}<br>
Fortunately expressions of the form:
{{{x^2 +bx + c}}} are fairly simple to factor. We just look for the factors of "c" that add up to "b". So in your equation we look for a "p" whose factors add up to -12 and whose factors are 2 apart from each other. With some thought and/or trial and error we should be able to find that -5 and -7 are two apart from each other and they add up to -12. Since "p" is the product of these two factors, "p" is (-5)(-7) = 35. The final equation is:
{{{x^2 -12x + 35 = 0}}} whose roots are (-5) and (-7).