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 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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