To find the minimum value of the quadratic function $f(x) = 5x^2 - 14x + 7$, we can use the formula for the vertex of a parabola. For a quadratic function in the form $ax^2 + bx + c$, the x-coordinate of the vertex is given by:

$x = \frac{-b}{2a}$

Here, $a = 5$, $b = -14$, and $c = 7$. Plugging in the values:

$x = \frac{-(-14)}{2(5)} = \frac{14}{10} = 1.4$

Now, substitute $x = 1.4$ back into the function to find the minimum value:

$f(1.4) = 5(1.4)^2 - 14(1.4) + 7$

$f(1.4) = 5(1.96) - 19.6 + 7$

$f(1.4) = 9.8 - 19.6 + 7 = -2.8$

Thus, the minimum value of the function is $-2.8$.

Do you want more details or have any questions?

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Here are 5 related questions:

1. What is the significance of the vertex in a quadratic function?
2. How do you determine whether a quadratic function opens upwards or downwards?
3. Can you find the roots of the equation $5x^2 - 14x + 7 = 0$?
4. How is the discriminant used to analyze quadratic equations?
5. How do you complete the square to rewrite a quadratic equation?

Tip: The vertex of a parabola is always the maximum or minimum point depending on the direction it opens (up or down).