1. Determine the parameter $m$ so that the graph of the quadratic function $f(x)=x^{2}+m x+3$ will touch the line $y=2$. Find all solutions.
1. We determine that the vertex of the quadratic function lies on the line $y=2$, so $q=2$. We use the formula for $D$ and $q$, substituting the given data $2=-\frac{m^{2}-12}{4}$, eliminate the fraction, and obtain $m^{2}-4=0$. The solutions to the resulting equation are $m_{1}=2$ and $m_{2}=-2$.

![](https://cdn.mathpix.com/cropped/2024_06_07_93a0a0489a8a740fd9e9g-11.jpg?height=63&width=1642&top_left_y=485&top_left_x=204)

Calculation of $D=m^{2}-12$..................................................................................................

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![](https://cdn.mathpix.com/cropped/2024_06_07_93a0a0489a8a740fd9e9g-11.jpg?height=54&width=1639&top_left_y=641&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_93a0a0489a8a740fd9e9g-11.jpg?height=60&width=1639&top_left_y=690&top_left_x=206)