The maximum a!!
Algebra
Level
5
Find the maximum value of , if
has at least one real root.
The answer is 6.333.
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1 solution
Let's say x^2 +x +1 = y, so x^2 +x+2 = y+1 ( I just did it in order to make the expression look easier). After replacing, we have :
(y+1)^2 - y(y+1)(a-3) + y^2(a-4) = 0
After simplifying, what we get is :
5y - ya + 1 = 0
Now, let's return our big expressions:
5(x^2 +x +1) - a(x^2 + x +1) + 1 = 0
You finally should get quadratic equation:
(5-a)x^2 + x(5-a) + (6-a) = 0
Now, according to conditions, the equation has to have at least one root. So, the D value ( b^2 - 4ac) has to be greater or equal than 0 ( if it is equal, we have one root, if it is greater, we have 2 roots):
D = (5-a)^2 - 4(6-a)(5-a) >= 0
This leads us to a quadratic inequality :
-3a^2 +34a - 95 >= 0
After solving inequality, you get [5 ; 6.333]. So, the maximum value of a is 6.333.
What I did is simple replacing. It is a very powerful technique.
SOLVED