Problem 70 - 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 \(f(x)=0.\) $$f(x)=-x^{2}+4 x-4$$

Step-by-Step Solution

Answer

The discriminant of the function \(f(x) = -x^{2}+4x-4\) is 0, which means it has one real root, thus, the graph of the function has just one x-intercept.

Step 1: Identifying Coefficients

Given the quadratic equation is \(f(x) = -x^{2}+4x-4\), one can identify that the coefficients of the equation are \(a = -1, b = 4, c = -4\). These are based on the standard form of the quadratic equation \(ax^2 + bx + c = 0\).

Step 2: Calculating the Discriminant

Next, plug the values of a, b, and c into the formula for the discriminant which is \(D = b^2 - 4ac\). This gives \(D = (4)^2 - 4(-1)(-4) = 16 - 16 = 0\).

Step 3: Interpreting the Discriminant

There are three interpretations for the discriminant (D): D>0 means 2 real roots and D=0 means 1 real root, and D<0 means no real roots. Because in our case D=0, it follows that the quadratic equation \(f(x) = 0\) has one real root, which also means the graph of the function f(x) has one x-intercept.

Key Concepts

Discriminant

The discriminant is a key component in determining the nature of the roots of a quadratic equation. For a quadratic equation in the form \( ax^2 + bx + c = 0 \), the discriminant \( D \) is calculated using the formula: \[ D = b^2 - 4ac \]

• If \( D > 0 \), the equation has two distinct real roots.
• If \( D = 0 \), there is exactly one real root, or a repeated root.
• If \( D < 0 \), the equation has no real roots but rather two complex roots, which aren't real numbers.

Understanding the discriminant helps us predict the number of solutions without actually solving the equation. In the given exercise, the discriminant \( D \) was calculated to be zero, indicating that there is one real solution and only one x-intercept on the graph.

Real Roots

Real roots indicate where the graph of a quadratic function crosses the x-axis. In simpler terms, they're the solutions to the equation \( ax^2 + bx + c = 0 \) when plotted on a graph. For our specific quadratic example,

• The discriminant \( D = b^2 - 4ac \) helps determine how many real roots exist.
• If \( D = 0 \), there's exactly one real root.

The presence of real roots is essential for graphing because they show the points where the function reaches the x-axis and change sign. In our exercise, the discriminant being zero tells us there's one real root, which translates into a lone point on the x-axis. This particular root can be found using the quadratic formula, although here it's understood from the discriminant analysis.

Quadratic Formula

The quadratic formula is a universal tool for finding the roots of any quadratic equation: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here's how it works:

• "\(+\sqrt{b^2 - 4ac}\)" and "\(\-\sqrt{b^2 - 4ac}\)" indicate that there can be two solutions for the quadratic equation, given that \( D > 0 \).
• If \( D = 0 \), then \( \sqrt{b^2 - 4ac} = 0 \), meaning both "+" and "-" yield the same result, this giving us one root.
• If \( D < 0 \), the expression inside the square root becomes negative, resulting in complex solutions.

The quadratic formula not only enables the computation of real roots but also provides insight into their nature based on the discriminant. For the exercise provided, because the discriminant is zero, the formula will give a single root for the equation, confirming our calculations of the discriminant.