Question

determine the equation of the parabola with roots 2 + square root of 3 and 2- square root of 3 and passing through the point (2,5)

Answer 1

Linear Algebra to the rescue; We want to satisfy all the three conditions
I checked the answer should be f(x)=(-5/3)x^2+(20/3)x-5/3
We have;
\[\huge \matrix{ax^2+bx+c=0 \\ a(2+\sqrt{3})^2+b(2+\sqrt{3})+c=0 \\a(2-\sqrt{3})^2+b(2-\sqrt{3})+c=0 } \]
Which leads to
\[\huge \matrix{(2+\sqrt{3})&&(2+\sqrt{3})&&1&&0\\(2-\sqrt{3})&&(2-\sqrt{3})&&1&&0\\ 4&&2&&1&&5}\]

Row reducing it leads to
\[\huge \matrix{1&&0&&0&&-\frac{5}{3} \\ 0&&1&&0&&\frac{20}{3} \\ 0&&0&&1&&-\frac{5}{3}} \]

Which are our solutions for a,b and c so we ave;
\[\huge f[x]=-\frac{5}{3}x^2+\frac{20}{3}x-\frac{5}{3}\]
Any questions?

Answer 2

First one should be \[\huge 4a+2b+c=5\] sorry about that