Symmetrically So
Try to solve this equation without multiplying it out. Notice a symmetry and make a substitution to a new variable and you can quickly solve this quartic by turning it into a quadratic...
If you are trying to find the square root of $4+2\sqrt{3}$ (i.e. you are trying to find $x$ such that $x^2=4+2\sqrt{3}$) then you can write $x=a+b\sqrt{3}$ and then find $a$ and $b$ such that $(a+b\sqrt{3})^2=4+2\sqrt{3}$.
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Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?
Good Approximations
Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.
Target Six
Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.