1975 AHSME Problems/Problem 27
Problem
If and are distinct roots of , then equals
Solution 1
If is a root of , then , or Similarly, , and , so
By Vieta's formulas, , , and . Squaring the equation , we get Subtracting , we get
Therefore, . The answer is .
Solution 2(Faster)
We know that . By Vieta's formulas, ,, and . So if we can find , we are done. Notice that , so , which means that
~pfalcon
Solution 3 (Beginner's Solution)
Use Vieta's formulas to get , , and .
Square , and get
Substitute and simplify to get
After that, multiply both sides by , to get
Then, factor out , , and :
Then, substitute the first equation into , , and .
Then, multiply it out:
After that, substitute the equations and :
Solving that, you get ~EZ PZ Ms.Lemon SQUEEZY