MHB Find equation of a quadratic with line of symmetry

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The axis of symmetry for a quadratic equation is given by x = -b/(2a). If an x-intercept is at (-2, 0), it leads to the equation 0 = a(-2)^2 + b(-2) + c, establishing relationships between b and c as functions of a. When the axis of symmetry is x = 1.75, the quadratic can be expressed as y = a(x - 1.75)^2 + c, resulting in b = 3.5a. Additionally, with the condition y = 0 when x = -2, it simplifies to 0 = 4a - 2b + 22, yielding b = 11 + 2a. This discussion highlights the interdependence of coefficients in forming a quadratic equation based on its symmetry and intercepts.
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Here's something to start you off. The axis of symmetry for the parabola [math]y = ax^2 + bx + c[/math] is [math]x = -\dfrac{b}{2a}[/math] and if an x-intercept is at (-2, 0) then [math]0 = a(-2)^2 + b(-2) + c[/math]. That will give you both b and c as functions of a.

Can you finish?

-Dan
 
Equivalently, if the axis of symmetry is x= 1.75 then the equation can be written as $y= a(x- 1.75)^2+ c= ax^2+ 3.5ax+ 3.0625+ c= ax^2+ bx+ 22. We must have b= 3.5a.

If, in addition, y= 0 when x= -2, 0= 4a- 2b+ 22 so b= 11+ 2a.
 

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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