Let's follow each step of the instructions. We start with a square and the midpoint of CD.
Next we draw the arc and the extension of CD. Then we find the intersection point Q.
Finally, we extend AB and draw a perpendicular from Q to DQ to construct the rectangle ARQD.
b
We are given that AD=x, and we know that ABCD is a square and ARQD is a rectangle. This means that we have several congruent segments on the figure all with measure x. Let's mark these on the figure.
We are also given that CQ=1.
Let's express the length of the segments involved in the given equation in terms of x.
DQ&=x+1
AD&=x
QR&=x
CQ&=1Let's substitute these expressions and rearrange the resulting equation.
We are asked to use the method of completing the square to solve this equation.
This method involves creating a complete square on the left hand side of the equation by adding the square of half the coefficient of the linear term to each side.
x^2+ bx=d
⇕
x^2+ bx+(b/2)^2=d+(b/2)^2
⇕
(x+b/2)^2=d+(b/2)^2
In our case b= -1, so we add ( -12)^2= 14 to each side.
