Question

What are the \(x\) -intercepts of \(y=x^{2}-2 x-3 ?\) \(\begin{array}{lllll}\text { (A) } 1 \text { and }-3 & \text { (B) }-2 \text { and }-3 & \text { (C) } 6 \text { and }-1 & \text { (D) } 3 \text { and }-1\end{array}\)

Short answer

The x-intercepts of the function are 3 and -1, matching the options in answer (D).

Step-by-step solution

Identifying values of a, b, c

In the equation \(y = x^{2} - 2x - 3\), a = 1, b = -2 and c = -3.

Apply the quadratic formula

Now, substitute the values of a, b, and c into the quadratic formula: \(x\) = \([-(-2) ± \sqrt{(-2)^{2} - 4(1)(-3)}] / 2(1)\), simplify it to get \(x = [2 ± \sqrt{4+12}]/2\), which further simplifies to \(x = [2 ± \sqrt{16}]/2\).

Calculate the x-intercepts

This results in two solutions: \(x = 3\) and \(x = -1\). These are the x-intercepts of the given function.

Key concepts

Quadratic Formula

If you've ever worked with quadratic functions, you've likely encountered the quadratic formula. It's a powerful tool for finding the x-intercepts of a quadratic equation of the form ax^2 + bx + c = 0. The formula is given by:

\[\begin{equation}x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\end{equation}\]
The symbols '±' mean that the quadratic equation has two solutions. To determine the x-intercepts of the function y = x^2 - 2x - 3, we set y to 0 and use the formula with a = 1, b = -2, and c = -3. After calculating the discriminant (b^2 - 4ac) and applying the square root, we get two possible values for x, which are the points where the graph of the quadratic function crosses the x-axis.

Factoring Quadratics

Factoring is another method to find the roots of a quadratic equation, which are the x-intercepts on the graph. The process involves rewriting the quadratic in the form (x - p)(x - q) = 0, where p and q are the solutions. For our example, y = x^2 - 2x - 3, we can factor the quadratic by looking for two numbers that multiply to give -3 (the c value) and add up to give -2 (the b value). Those numbers are -3 and +1.

\(x^2 - 2x - 3 = (x - 3)(x + 1)\)
Setting each factor equal to zero gives us the same x-intercepts, x = 3 and x = -1. Factoring quadratics is often quicker and less error-prone than using the quadratic formula, especially with simple equations.

Solving Quadratic Equations

Solving quadratic equations can be done using multiple methods, including the quadratic formula, factoring, graphing, and completing the square. For the equation y = x^2 - 2x - 3, we've already found that the solution using the quadratic formula yields x-intercepts of 3 and -1. It’s important to check the validity of solutions by substituting them back into the original equation.

Substituting x = 3, we get (3)^2 - 2(3) - 3 = 0, and for x = -1, (-1)^2 - 2(-1) - 3 = 0, confirming that both are true solutions. Understanding how to solve quadratic equations is essential in mathematics, as it not only allows for finding points where the graph crosses the x-axis but also solves various real-world problems.