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Chapter 2: Problem 93

Suppose \(f\) is a quadratic function such that the equation \(f(x)=0\) has two real solutions. Show that the average of these two solutions is the first coordinate of the vertex of the graph of \(f\).

Short Answer

Expert verified
The average of the two real solutions (roots) of a quadratic function \(f(x)=ax^2 + bx + c\) is given by \(\frac{-b}{a}\). The x-coordinate of the vertex of the quadratic function is given by \(\frac{-b}{2a}\). Comparing these two results, we find that they are equal, so the average of the roots is equal to the x-coordinate of the vertex.

Step by step solution

01

A general quadratic function can be written as: \[f(x) = ax^2 + bx + c\] where a, b, and c are constants. To find the real solutions (roots) of the quadratic equation \(f(x)=0\), we use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] #Step 2: Calculate the average of the roots using the quadratic formula#

To find the average of the roots, we first need to find the sum of the roots using the quadratic formula. Adding both roots, we get: \[x_1 + x_2 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} + \frac{-b - \sqrt{b^2 - 4ac}}{2a}\] Then, we can simplify this expression: \[x_1 + x_2 = \frac{-b + \sqrt{b^2 - 4ac} - b - \sqrt{b^2 - 4ac}}{2a} = \frac{-2b}{2a}\] Now, we calculate the average of the roots: \[\text{Average} = \frac{x_1 + x_2}{2} = \frac{\frac{-2b}{2a}}{2} = \frac{-b}{a}\] #Step 3: Write the formula for the x-coordinate of the vertex of a quadratic function#
02

The x-coordinate of the vertex of a quadratic function is given by the formula: \[x_{vertex} = \frac{-b}{2a}\] #Step 4: Compare the average of the roots and the x-coordinate of the vertex#

From Step 2, we found that the average of the roots is \(\frac{-b}{a}\) and from Step 3, we found that the x-coordinate of the vertex is \(\frac{-b}{2a}\). Comparing these two results, we see that: \[ \frac{-b}{a} = \frac{-b}{2a}\] Thus, the average of the two real solutions (roots) of the quadratic function is equal to the first coordinate (x-coordinate) of the vertex of the graph of \(f\).

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