Geometry Parabola $2x^2+\alpha x+3\alpha$ to find common point

Question

Can you help me find the answer to this question?

For any real number $\alpha$, the parabola $f_{\alpha}(x) = 2x^2 + \alpha x + 3\alpha$ passes through the common point $(a, b)$. What is the value of $a + b$?

Answer 1

So for any $\,\alpha\in\Bbb R\,$ ,we have that

$$b=f_\alpha(a)=2a^2+a\alpha+3\alpha\Longrightarrow $$

Since this is true for any $\,\alpha\in\Bbb R\,$ , let us choose:

$$\;\;\;\;\;\;\;\;\;\;\;\;\;\begin{align*}(1)\;\;\;\alpha=0:& \,\,b=2a^2\\(2)\;\;\;\alpha=1:&\,\,b=2a^2+a+3\end{align*}$$

Comparing (1)-(2), we get

$$a+3=0\Longrightarrow a=-3\Longrightarrow b=2\cdot 3^2=18\Longrightarrow a+b=15$$

Answer 2

Choose two values for $\alpha=0,1$ and set $f_0(x)=f_1(x)$ to get: $$ 2x^2=2x^2+x+3 \\ x+3=0 \rightarrow x=-3 \rightarrow y=2(-3)^2=18 $$ Then $x+y=-3+18=15$.