SOLUTION: For what values of p will one root of {{{p(x^2+1)=8x-1}}} be three times the other root?

Question 228262: For what values of p will one root of be three times the other root?
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Let's first look at a plan. Let's call one of the roots, r. Then the other root, being three times as much, would be 3r. And an equation which has r and 3r as roots would be . (Think this out and make sure you see why, if x was r or 3r, then it would fit this equation.) If we simplify this equation we get:

Our plan then is to rearrange your equation into this form and then figure out what "P" would have to be. So we'll start by dividing both sides by p:

Next we'll subtract form each side:

Now we'll add
We want this equation

to match the pattern of

This must mean that
so the coefficients of x match
and
so the constant terms match.
We now have a system of two equations with two variables. We should be able to solve this system. Since the equations are not linear, probably the Substitution Method is best. So we'll start by solving one of the equations for one of the variables. It looks to me that solving the first equation for r is easiest. We just have to divide both sides of the equation by 4:

Now we'll substitute this into the second equation:

And solve it:

Multiply both sides by the Lowest Common Denominator (lcd) to get rid of the fractions:

This is a quadratic equation to we'll get one side equal to zero:

and factor it:

and solve it:
or
or

You can check these answers, one at a time, by substituting the value for p into your original equation and finding its roots. You should find that for each value of p that one root is three times the other. I'll leave this up to you.
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