Question

For a quadratic equation \(a x^2+b x+c=0\) with two real solutions, show that the mean of the solutions is \(-\frac{b}{2 a}\). How is this fact related to the symmetry of the graph of \(y=a x^2+b x+c ?\)

Short answer

The mean of the roots of a quadratic equation \(ax^2 + bx + c = 0\) is \(-\frac{b}{2a}\) and this equivalent to the line of symmetry of the graph of the quadratic function.

Step-by-step solution

Identify the formula of solutions

According to the quadratic formula, the solutions of the equation \(ax^2 + bx + c = 0\) are given by \(x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Calculate the average of solutions

The mean (average) of the solutions is given by \(Mean = \frac{x_1 + x_2}{2} = \frac{\frac{-b + \sqrt{b^2 - 4ac}}{2a} + \frac{-b - \sqrt{b^2 - 4ac}}{2a}}{2}\). Simplifying, the mean equal to \(-\frac{b}{2a}\).

Referring to the graph of the quadratic equation

The quadratic function's graph is a parabola that is symmetric to the line \(x = -\frac{b}{2a}\), which is none other than the x-coordinate of the vertex of the parabola, which according to the symmetry of the graph, is the mean of the x-intercepts (solutions).

Key concepts

Solutions of Quadratic Equations

Quadratic equations have the standard form \(ax^2 + bx + c = 0\). These equations can have two real solutions, one real solution, or two complex solutions, depending on the discriminant \(b^2 - 4ac\). The solutions, also known as roots, can be found using various methods such as factoring, completing the square, or by applying the quadratic formula. In our example, we focus on equations with real solutions. For an equation to have real solutions, the discriminant \(b^2 - 4ac\) must be non-negative. This ensures the presence of two distinct or one repeated real root. Understanding the nature of these roots is crucial in solving and analyzing quadratic equations efficiently.

Mean of Solutions

The mean of the solutions of a quadratic equation is crucial to finding other properties of the equation. For the solutions \(x_1\) and \(x_2\) of the quadratic equation \(ax^2 + bx + c = 0\), the average of these solutions is expressed as \(\frac{x_1 + x_2}{2}\). By using the quadratic formula, \(x_1\) and \(x_2\) can be written as \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). By adding these two solutions and simplifying, we find the mean to be \(-\frac{b}{2a}\). This simple expression reveals a lot about the parabola's characteristics without needing the exact solutions.

Symmetry of Parabolas

Parabolas, which are the graphs of quadratic functions, have a symmetrical property. This means if you draw a vertical line through the vertex, both sides of the parabola mirror each other perfectly. For the quadratic equation \(y = ax^2 + bx + c\), this line of symmetry is given by \(x = -\frac{b}{2a}\). This is exactly the same as the mean of the solutions of the equation. The line \(x = -\frac{b}{2a}\) acts as a pivot, making the parabola symmetric about this line. Hence, the mean of the roots is crucial as it directly corresponds to the parabola's symmetry axis. This symmetry helps in predicting the behavior of the parabola more efficiently.

Quadratic Formula

The quadratic formula is a reliable method for finding solutions to any quadratic equation \(ax^2 + bx + c = 0\). It is given by \(x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula comes in handy when factoring is difficult or when the equation doesn't easily simplify. The formula uses the discriminant \(b^2 - 4ac\) to determine the nature of the solutions: if positive, there are two different real solutions; if zero, one real solution (a repeated root); and if negative, two complex solutions. The quadratic formula is powerful because it always leads you to the roots and does so using straightforward arithmetic operations.

Vertex of a Parabola

In a quadratic function \(y = ax^2 + bx + c\), the vertex is a key point that represents the maximum or minimum value of the function on its graph. The vertex coordinates \( (h, k) \) include the x-coordinate \(h = -\frac{b}{2a}\), which aligns with the mean of the roots and the axis of symmetry. The value \(k\) represents the function value when \(x = -\frac{b}{2a}\), hence solving \(y = a(-\frac{b}{2a})^2 + b(-\frac{b}{2a}) + c\) gives \(k\). By locating the vertex, you can easily determine the parabola's apex, guiding you to understand whether it's an upward or downward opening parabola and to discover other essential features like the direction the parabola faces and finding maximum/minimum values.