Problem 1 - Algebra and Trigonometry

Question

The solutions of the equation \(x^{2}-2 x-3=0\) are the _________ intercepts of the graph of \(y=x^{2}-2 x-3\)

Step-by-Step Solution

Answer

The solutions, or x-intercepts, are \(x = 3\) and \(x = -1\).

Step 1: Understand the Equation

The equation given is a quadratic equation, \(x^{2} - 2x - 3 = 0\), which defines a parabola when expressed as \(y = x^{2} - 2x - 3\). The solutions to the equation represent the points where the parabola intersects the x-axis, known as the x-intercepts.

Step 2: Use the Quadratic Formula

To find the solutions of the quadratic equation, apply the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\), where \(a=1\), \(b=-2\), and \(c=-3\).

Step 3: Calculate the Discriminant

Compute the discriminant of the quadratic equation: \(b^{2} - 4ac\). For our equation, it is \((-2)^{2} - 4 \cdot 1 \cdot (-3) = 4 + 12 = 16\).

Step 4: Find the Roots Using the Formula

Substitute the discriminant and other values into the quadratic formula: \(x = \frac{-(-2) \pm \sqrt{16}}{2 \times 1}\). Simplifying gives \(x = \frac{2 \pm 4}{2}\).

Step 5: Solve for x

Calculate the values of \(x\) by evaluating \(2 + 4\) and \(2 - 4\):\[x = \frac{6}{2} = 3\]and\[x = \frac{-2}{2} = -1\].

Step 6: Determine the x-Intercepts

The solutions \(x = 3\) and \(x = -1\) are the x-intercepts of the graph \(y = x^{2} - 2x - 3\), indicating where the parabola crosses the x-axis.

Key Concepts

Understanding X-Intercepts in a Quadratic Equation

In any quadratic equation, such as \(y = ax^2 + bx + c\), the x-intercepts are where the graph of the parabola crosses the x-axis. These points provide the solution to the equation when set to zero, as illustrated by \(x^2 - 2x - 3 = 0\).
The x-intercepts are the values of \(x\) where \(y\) becomes zero, which can be found by solving the quadratic equation. In our example, these occur at \(x = 3\) and \(x = -1\), representing the points \((3, 0)\) and \((-1, 0)\) on the graph. Identifying the x-intercepts helps in understanding the shape and position of parabola graphs on a coordinate plane. The parabola can have zero, one, or two x-intercepts, depending on the discriminant's value.

The Quadratic Formula: A Key Tool

The quadratic formula is an essential tool used to find the roots of any quadratic equation. This formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is universally applicable, providing a systematic way to determine the solutions, or x-intercepts.
In the context of the equation \(x^2 - 2x - 3 = 0\), the formula helps find where the parabola crosses the x-axis. By plugging the coefficients \(a = 1\), \(b = -2\), and \(c = -3\) into the formula, we can simplify and solve for \(x\), yielding \(x = 3\) and \(x = -1\). This method is both efficient and reliable, ensuring precision in finding exact values.

Role of the Discriminant

The discriminant, represented as \(b^2 - 4ac\), plays a crucial role in understanding the nature of a quadratic equation's roots. Its value determines the number and type of solutions, or x-intercepts.
For the equation \(x^2 - 2x - 3 = 0\), the discriminant is calculated as \((-2)^2 - 4 \times 1 \times (-3)\), resulting in \(16\). A positive discriminant like 16 indicates two distinct real roots, which means the parabola will intersect the x-axis at two points. Conversely, a zero discriminant would mean one repeated root, and a negative discriminant would imply no real x-intercepts.

Characteristics of a Parabola

A parabola is a symmetrical curve that represents the graph of a quadratic equation. Expressed in the form \(y = ax^2 + bx + c\), it always opens upwards or downwards depending on the sign of \(a\). If \(a > 0\), it opens upwards, and if \(a < 0\), it opens downwards.
For our specific equation, \(y = x^2 - 2x - 3\), the parabola opens upwards as \(a = 1\). The vertex form can help pinpoint its turning point, providing a deeper understanding of its shape. Understanding parabolas is vital for graphically solving quadratic equations and interpreting real-world scenarios involving quadratic relationships. This conical shape appears in various fields, from physics to architecture.