Problem 67 - Introductory Algebra for College Students

Question

Replace \(y\) with 0 and find the \(x\) -intercepts for the graph of \(y=x^{2}-2 x-3\)

Step-by-Step Solution

Answer

The x-intercepts of the graph are \(x = 3\) and \(x = -1\).

Step 1: Set equation equal to zero

To find the x-intercepts, replace \(y\) with 0 in the equation. This leads to \(0 = x^{2} - 2x - 3\). Solving this equation will give us the x-values of the intercepts.

Step 2: Apply the Quadratic Formula

Use the quadratic formula, which is \(x = [-b \pm \sqrt{b^{2}-4ac}]/(2a)\) for an equation of the form \(ax^2 + bx + c = 0\). In the equation \(0 = x^{2} - 2x - 3\), \(a = 1\), \(b = -2\), and \(c = -3\). Putting these values into the quadratic formula, we have: \(x = [2 \pm \sqrt{(-2)^{2}-4*1*(-3)}]/(2*1)\), which simplifies to: \(x = [2 \pm \sqrt{4 + 12}]/2 \).

Step 3: Simplify the expression

Solve under the square root: \(x = [2 \pm \sqrt{16}]/2\), which further simplifies to \(x = [2 \pm 4]/2\). Thus we have two solutions: \(x = 6/2 = 3\) and \(x = -2/2 = -1\).

Step 4: Check the solution

Finally, substitute the obtained x-values into the original equation to ensure they satisfy \(y = 0\). When \(x = 3\), substitute back into the equation \(y = x^{2} - 2x - 3\) to get \(y = (3^{2}) - 2*3 - 3 = 9 - 6 - 3 = 0\). The same steps can be followed when \(x = -1\) to see that it also satisfies \(y = 0\). Thus both solutions are correct.

Key Concepts

Quadratic Formula

When faced with a quadratic equation, such as finding the x-intercepts for a graph, the quadratic formula is a powerful tool. The formula, given by \(x = [-b \pm \sqrt{b^{2}-4ac}]/(2a)\), offers a solution for any quadratic equation of the standard form \(ax^2 + bx + c = 0\). Here's how it works:

First, identify the coefficients: \(a\), \(b\), and \(c\) from your quadratic equation. These correspond to the coefficient of \(x^2\), the coefficient of \(x\), and the constant term, respectively. Once identified, simply plug these values into the quadratic formula. The term under the square root, \(b^{2}-4ac\), is known as the discriminant, and it is crucial as it determines the nature and number of solutions.

After inserting the coefficients, you will have to deal with the plus-minus \(\pm\) symbol. This means you will have two values to calculate: one for the addition and another for the subtraction. Both of these potential results are the solutions for \(x\) and represent the x-intercepts of the graph in a geometric context. In our exercise, by substituting \(a = 1\), \(b = -2\), and \(c = -3\) into the formula, we observe how simplification leads to finding the precise intercepts.

Quadratic Equation

Understanding the structure of a quadratic equation is equally essential. In the general form, \(ax^2 + bx + c = 0\), the equation represents a parabola when plotted on a coordinate grid. The highest exponent in a quadratic equation is two, which signifies that there will be a squared variable, and this is what gives the graph its distinct 'U' shape.

The x-intercepts of a quadratic equation (also known as the roots, zeros, or solutions) are the points where the graph crosses the x-axis, indicating where the value of \(y\) is zero. To locate these intercepts algebraically, we solve the quadratic equation. Depending on the value of the discriminant, a quadratic equation might have two distinct real solutions, one repeated real solution, or no real solutions but rather two complex solutions. Our example, where \(y = x^2 - 2x - 3\), showcases a parabola that intersects the x-axis at two points, exhibiting two distinct real solutions for the x-intercepts.

Simplifying Expressions

Once we apply the quadratic formula, we arrive at an expression that often requires simplification to reach the final solution. Simplification is the process of making an algebraic expression as straightforward as possible. This often involves combining like terms, reducing fractions, and performing arithmetic operations such as addition, subtraction, multiplication, or division.

For example, taking the step \(x = [2 \pm \sqrt{16}]/2\), we see that \(\sqrt{16} = 4\), and the expression simplifies to \(x = (2 \pm 4)/2\). Splitting this into two separate calculations, we get \(x = (2 + 4)/2\), which simplifies to \(x = 3\), and \(x = (2 - 4)/2\), which simplifies to \(x = -1\). Both results are the x-intercepts of the graph. Simplifying expressions is a critical step in the process, ensuring that we arrive at the most accessible form of the solution and can readily interpret the meanings of our results.