MHB Finding the Minimum Value of a Parabola: Tips and Techniques

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To find the minimum value of the quadratic function f(x)=3x^2+10x+219, the vertex formula x=-b/(2a) can be used, where a=3 and b=10. Calculating this gives x=-10/(2*3)=-5/3. Substituting x back into the function provides the minimum value of f(x). Additionally, participants are reminded to adhere to forum rules, particularly regarding clarity in expressing difficulties. Understanding the vertex's role is crucial for solving problems involving parabolas.
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What is the smallest value in the image set of the function:

f(x)=3x^2+10x+219
 
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Welcome to the forum!

The vertex (minimum or maximum) of a parabola $f(x)=ax^2+bx+c$ has $x$-coordinate $x=-\dfrac{b}{2a}$.

For the future, please read the forum http://mathhelpboards.com/rules/, especially rule #11. In this case, you could write what facts about parabolas you know and what exactly your difficulty is in applying them.
 

Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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