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For how many values of $a$ is it true that the line $y=x+a^{2}-6$ passes through the vertex of the parabola $y=4x^{2}-8x+a^{2}$? (Adapted From 2005 AMC 12B Problem, Question \#8) 
Overseas Competition->Knowledge Point->Calculation Modules
We see that the vertex of the quadratic function $y=4x^{2}-8x+a^{2}$ is $\left(1, a^{2}-4\right)$. If $\left(1, a^{2}-4\right)$ will be on the line $y=x+a^{2}-6$, $a^{2} -4=1+a^{2}-6$. Solve for $a$, there is no solution. 