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A quadratic function is a polynomial function of degree 2, typically expressed in the form \[f(x) = ax^2 + bx + c\] where 'a', 'b', and 'c' are constants, and 'x' represents the variable.
The graph of a quadratic function is a parabola, which opens upwards if 'a' is positive and downwards if 'a' is negative.
The vertex of the parabola represents either the maximum or minimum point of the function.
In our example, the quadratic function given is \[f(x) = 14 - x^2\]. Here, 'a' is negative (-1), causing the parabola to open downwards, making the vertex the maximum point.
Understanding the shape and properties of quadratic functions helps in solving related inequalities and interpreting solutions.

Solving quadratic inequalities often involves several steps:

• Setting up the inequality
• Isolating the quadratic term
• Solving for the variable
• Interpreting the solution

To start, we set up the given inequality \[f(x) > 5\] which translates to \[14 - x^2 > 5\].
Next, we isolate the quadratic term by subtracting 5 from both sides, resulting in \[9 - x^2 > 0\].
We simplify it to \[9 > x^2\], which can also be written as \[x^2 < 9\].
Taking the square root of both sides, we obtain the inequalities \[-3 < x < 3\]. This method can be applied to any quadratic inequality, making it a valuable tool.

Interpreting the solution of a quadratic inequality requires understanding the range of values that satisfy the inequality. We found that \[-3 < x < 3\].
This means any value of 'x' within this range will make \[f(x) > 5\].
It's important to note that the values -3 and 3 are not included because the inequality is strictly greater than 5 (\[f(x) > 5\], not \[f(x) \,\geq \, 5\]).
Therefore, we describe the solution using interval notation as \((-3, 3)\).
This notation is crucial in expressing the set of all possible 'x' values that satisfy the inequality.
Always carefully read and interpret the given inequality to determine the correct solution set.