Problem 65 - 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 8. Thus, the function will have two x-intercepts and there will be two real roots or solutions to the equation \(f(x) = 0\)

Step 1: Identify coefficient values

In the given quadratic function \(f(x) = x^{2} -2x -1\), the coefficients are \(a=1\), \(b=-2\), and \(c=-1\)

Step 2: Calculate the Discriminant

To find the discriminant, apply the formula, which is \(D = b^{2} - 4ac\). Substituting the identified coefficients, the discriminant becomes \(D = (-2)^{2} - 4 * 1 * (-1)\)

Step 3: Evaluate the Discriminant

Evaluating the formula gives \(D = 4 + 4 = 8\)

Step 4: Analyze the Discriminant

Since the discriminant is greater than 0, the equation \(f(x) = 0\) has 2 real roots which mean the graph of \(f(x)\) will cross the x-axis at two points

Step 5: Determine the number of real solutions

As determined through the analysis of the discriminant, there are 2 real solutions to the equation \(f(x) = 0\)

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of the second degree. It is expressed in the standard form as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The highest power of the variable \( x \) in a quadratic equation is two. This characteristic gives it the name "quadratic," which comes from "quad," meaning square.

To solve a quadratic equation, one often looks for values of \( x \) that make the equation true, known as the solutions or roots.

• The quadratic can graphically be represented as a parabola on a coordinate plane.
• The direction (upward or downward) of the parabola depends on the sign of \( a \). If \( a \) is positive, the parabola opens upwards. If \( a \) is negative, it opens downwards.

The vertex of the parabola (the highest or lowest point, depending on its direction) and the intersections with the x-axis (if any) are key features analyzed when solving or graphing quadratic equations.

Real Roots

Real roots of a quadratic equation are the solutions that are real numbers and correspond to the points where the graph of the equation touches or crosses the x-axis. The number and type of roots depend on the discriminant, which is a component of the quadratic formula \( b^2 - 4ac \).

There are three possibilities:

• If the discriminant \( D \) is greater than zero, the quadratic equation has two distinct real roots. This indicates two x-intercepts on the graph.
• If \( D \) equals zero, it results in one real root, indicating the parabola touches the x-axis at a single point—a repeated root.
• If \( D \) is less than zero, the quadratic equation has no real roots and consequently has no x-intercepts. This means the parabola does not cross the x-axis.

In our example with \( f(x) = x^2 - 2x - 1 \), calculating the discriminant revealed \( D = 8 \), which means there are two distinct real roots.

Number of Solutions

The number of solutions of a quadratic equation directly relates to the nature of its roots. While calculating the discriminant, one can determine how many times the parabola will intersect the x-axis. This count of x-intercepts provides the number of real solutions for the equation:

• When \( D = 0 \), there is exactly one solution. The quadratic's graph grazes the x-axis at the vertex, and the solution is a repeated root.
• When \( D > 0 \), there are two solutions, as the parabola intersects the x-axis at two distinct points. This condition indicates two distinct real roots.
• When \( D < 0 \), no solution exists in real numbers, as the parabola does not touch the x-axis. The roots here would be complex or imaginary.

Thus, in the example given, with a calculated discriminant of 8 ensuring \( D > 0 \), we confirm there are two real solutions. This means the quadratic equation has two points where its graph intersects the x-axis, clearly establishing the real roots from our earlier discussion.