Problem 93 - Algebra 1: Concepts and Skills

Question

Use a graph to estimate the solutions of the equation. Check your solutions algebraically. (Lesson 9.5) $$ x^{2}-2 x-3=0 $$

Step-by-Step Solution

Answer

The values from the quadratic formula confirm the solutions estimated from the graph. The roots of the equation are \(x = -1\) and \(x = 3\).

Step 1: Graph the Equation

First, sketch the graph of the given function \(y = x^{2} - 2x - 3\). The graph is a parabola that opens upwards because the coefficient of \(x^{2}\) is positive. The solutions to the equation \(x^{2}-2x-3=0\) will be the x-coordinate of the points where the graph intersects the x-axis.

Step 2: Estimate Solutions from the Graph

After drawing the graph, estimate the points where the graph intersects the x-axis. These values of x will be the estimated solutions to the equation.

Step 3: Confirm the Solutions Algebraically

From the graph, we can estimate solutions, but to check the accuracy, apply the quadratic formula, which is given by \(x = (-b \pm \sqrt{b^{2} - 4ac}) / 2a\). Here, \(a = 1\), \(b = -2\), and \(c = -3\). Substitute these values into the formula and simplify to find the exact solutions.

Key Concepts

Parabola

The term "parabola" often comes up in the study of quadratic equations. A parabola is a smooth, U-shaped curve that graphs the quadratic function. It can open upwards or downwards, depending on the sign of the coefficient in front of the squared term. For the function \( y = x^2 - 2x - 3 \), the parabola opens upwards because the coefficient of \( x^2 \) is positive.
Parabolas have unique features:

• **Vertex**: The vertex is the point where the parabola turns. For upward-opening parabolas, this point is the lowest on the graph, and for downward-opening parabolas, it's the highest.
• **Axis of Symmetry**: This is a vertical line that divides the parabola into two mirror-image halves. For our equation, it's \( x = 1 \).
• **Intercepts**: The parabola may cross the x-axis at points known as the "roots" or "solutions" of the quadratic equation. It intersects the y-axis at \( y \) when \( x = 0 \).

Understanding these characteristics helps when graphing and solving quadratic equations.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation. It's expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In this formula, \( a \), \( b \), and \( c \) are coefficients from the quadratic equation in the form \( ax^2 + bx + c = 0 \). For the equation \( x^2 - 2x - 3 = 0 \), the values are \( a = 1 \), \( b = -2 \), and \( c = -3 \).
Using the quadratic formula:

• Substitute the values into the formula: \( a = 1, b = -2, c = -3 \).
• Calculate the discriminant \( b^2 - 4ac \). If it's positive, there are two solutions; if zero, one solution; negative, no real solutions.
• Solve for \( x \) by performing the arithmetic.

This algebraic method provides the exact solutions to the quadratic equation, unlike estimations from graphs.

Graphing

Graphing a quadratic equation visually represents solutions and helps in understanding its properties. Here's a simplified method to graph \( y = x^2 - 2x - 3 \):
Start by identifying the vertex of the parabola. Use the formula \( x = \frac{-b}{2a} \) to find the x-coordinate of the vertex.

• Calculate: \( x = \frac{-(-2)}{2(1)} = 1 \).
• Substitute \( x = 1 \) back into the equation to get the y-coordinate: \( y = 1^2 - 2(1) - 3 = -4 \). Thus, the vertex is \( (1, -4) \).
• Plot the vertex and then determine where the function intersects the y-axis when \( x = 0 \). The y-intercept is \( (0, -3) \).

Once these points are plotted on the graph, draw a symmetrical curve through these points, noting that it opens upwards. Estimating solutions graphically involves finding where the parabola crosses the x-axis. These crossing points are rough estimates of the solutions, giving an intuitive sense of where the solutions lie on the number line.