what is the minimum value of the function 5x^2-14x+7

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 \( -2.8 \).

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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).

Discover how to find the minimum value of the quadratic function 5x^2 - 14x + 7 using the vertex formula. This detailed solution includes step-by-step calculations and concepts such as algebra, parabolas, and quadratic functions, suitable for students in Grades 9-11.

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