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Understanding how to factor quadratic equations is crucial for solving them efficiently. Factoring is the process of breaking down the equation into simpler terms that are multiplied together to give the original equation. In the context of the given exercise, factoring involves finding two binomials that multiply together to give the original quadratic function, \(f(x)=x^{2}-8x+15\).

To approach this, we look for two numbers that multiply to give the constant term, in our case 15, and also add up to give the coefficient of the middle term, which is -8 in our example. These numbers are -3 and -5, and factoring by these yields \((x-3)(x-5)=0\). It’s like working on a puzzle, except the pieces are numbers and mathematical operations that need to fit together perfectly.

Once a quadratic equation has been factored, finding its roots becomes a straightforward task. The roots, also known as zeros or x-intercepts, are the values of x that make the function equal to zero. From the factored form, in this case, \((x-3)(x-5)=0\), we set each factor equal to zero—this is based on the principle that if two factors multiply to zero, at least one of the factors must be zero.

So, we solve \(x-3=0\) to find \(x=3\) and \(x-5=0\) to find \(x=5\), revealing the roots of our quadratic function. These solutions represent where the graph of the quadratic equation would cross the x-axis, which provides a visual understanding of the roots’ significance in a function’s graph.

Analyzing quadratic functions involves more than just finding the roots. It includes understanding the function’s graph, the vertex (the highest or lowest point), the axis of symmetry (the vertical line that divides the graph into two mirror images), and the direction of the parabola (upwards or downwards).

For the function \(f(x)=x^{2}-8x+15\), the vertex can be found using the formula \(-\frac{b}{2a}\). Here, \(a=1\) and \(b=-8\), so the vertex's x-coordinate is 4. Plugging this value back into the function provides the y-coordinate of the vertex. The axis of symmetry is the line \(x=4\), and since \(a=1\) is positive, the parabola opens upwards. These features combined give a complete picture of how the quadratic function behaves, beyond just its roots.