Problem 69 - Precalculus : Building Concepts and Connections

Question

For function of the form \(f(x)=\) \(a x^{2}+b x+c,\) find the discriminant, \(b^{2}-4 a c,\) and use it to determine the number of \(x\)-intercepts of the graph of \(f .\) Also determine the number of real solutions of the equation \(f(x)=0.\) $$f(x)=x^{2}+2 x+1$$

Step-by-Step Solution

Answer

The discriminant of the function \(f(x)=x^{2}+2x+1\) is 0. This means that there is exactly one real root of the equation \(f(x)=0\), and therefore there is exactly one x-intercept point on the graph of the function.

Step 1: Identify coefficients

The given function is \(f(x)=x^{2}+2x+1\). From this, the coefficients can be identified: a=1 (coefficient of \(x^{2}\)), b=2 (coefficient of x), and c=1 (the constant).

Step 2: Calculate the discriminant

The discriminant, \(D\), of a quadratic equation of the form \(ax^{2}+bx+c=0\) is given by the formula \(D=b^{2}-4ac\). Substituting the identified values of a, b, and c from the first step into this formula, we have \(D=(2)^{2}-4(1)(1)=4-4=0\). Thus, the discriminant D equals 0.

Step 3: Analyze the discriminant

The discriminant (_D_) provides information about the roots of the quadratic equation. If _D_ > 0, the equation has two distinct real roots. If _D_ = 0, the equation has one real root or a repeated root. If _D_ < 0, the equation has no real roots, only complex roots. In this case, since _D_ = 0, it means the quadratic equation \(f(x)=0\) has one real root or else a repeated root.

Step 4: Conclusion

Considering the graph of this function, the x-intercepts are the x-values where the function equals zero. Since the discriminant equals zero, there is one real root, hence there is only one x-intercept point on the graph of this function. The number of real solutions of the equation \(f(x)=0\) is also one because the discriminant is zero.

Key Concepts

Quadratic Equation

A quadratic equation is an expression of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. The equation represents a parabola when graphed on a coordinate plane. The solutions to the quadratic equation are those values of x that satisfy the equation, which can be found using various methods including factoring, completing the square, and the quadratic formula. The discriminant, denoted by D, is a key component calculated from the coefficients of the quadratic equation (D = b^2 - 4ac). It gives us vital information regarding the nature and number of solutions.

For a better understanding, let's add some practical insight into solving a quadratic equation. Using the example from the textbook f(x) = x^2 + 2x + 1, identify the coefficients first: a=1, b=2, and c=1. The discriminant tells us whether the solutions are real numbers, and whether they are distinct or repeated. The quadratic formula, x = (-b ± √D) / (2a), incorporates the discriminant, providing the exact solutions when they exist.

X-Intercepts

The x-intercepts of a graph are the points where the graph crosses the x-axis, and are important in understanding the behavior of the function represented by the graph. For a quadratic function, these intercepts correspond to the real solutions of the equation f(x) = 0. In other words, they are the x-values where the quadratic curve touches or intersects the x-axis.

Let's delve into the example of f(x) = x^2 + 2x + 1. After calculating the discriminant (step 2 in the solution) and finding it to be zero, it indicates that our function has a single x-intercept. This x-intercept is also the vertex of the parabola, where the graph just touches the x-axis and turns around. It's valuable to visualize that a discriminant of zero means the parabola sits on the axis at one distinct point, which is the case with our function.

Real Solutions

The term 'real solutions' refers to the answers of the quadratic equation that are real numbers, as opposed to complex or imaginary numbers. Real solutions are the x-values where the quadratic equation produces a zero value when evaluated. f(x) = 0 represents the condition where the graph of the quadratic function intersects the x-axis.

Considering the exercise at hand, with the discriminant being zero, it is clear that the equation f(x) = x^2 + 2x + 1 has a single real solution. This solution is the point where the parabola's vertex touches the x-axis. The fact that there's only one solution points toward the concept that the parabola is either perfectly sitting on the axis (one intercept) or not crossing it at all (zero intercepts, which happens when the discriminant is negative). Hence, the discriminant is a powerful tool for predicting the number of real solutions without necessarily computing the solutions explicitly.