SOLUTION: Find the relationship between p and q, if the equation px² + 3qx + 9 = 0 has equal roots.

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Question 1177574: Find the relationship between p and q, if the equation px² + 3qx + 9 = 0 has equal roots.
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
+px%5E2+%2B+3qx+%2B+9+=+0

Divide through by p

+x%5E2+%2B+expr%283q%2Fp%29x+%2B+9%2Fp+=+0

The sum of the roots is the coefficient of x with the opposite sign, -3q/p.
Since the roots are equal, let them be r and r.

-3q%2Fp=r%2Br
-3q%2Fp=2r

The product of the roots is the constant term 9/p.
Since the roots are both r, their product is r2.

9%2Fp=r%5E2

So we have a system of equations to solve

system%28-3q%2Fp=2r%2C9%2Fp=r%5E2%29

Solve the first for r

-3q%2F%282p%29=r

Substitute in the 2nd equation in the system:

9%2Fp=%28%28-3q%29%2F%282p%29%29%5E2

9%2Fp=%289q%5E2%29%2F%284p%5E2%29

Cross-multiply:

36p%5E2=9pq%5E2

4p%5E2=pq%5E2

p cannot be 0, so we may divide both sides by it

4p=q%5E2

Edwin