Problem 11 - Introductory Algebra for College Students

Question

Use factoring to solve each quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\) -intercepts. $$x^{2}-2 x-15=0$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation \(x^{2}-2 x-15=0\) are \(x=5\) and \(x=-3\).

Step 1: Factor the Quadratic Equation

In order to factor the quadratic equation \(x^{2}-2 x-15=0\), it's necessary to find two integers that multiply to the constant term -15 and add to the coefficient -2. The numbers -5 and 3 meet these conditions, as \(-5 * 3 = -15\) and \(-5 + 3 = -2\). Hence, the factored form is \( (x-5)(x+3)=0\).

Step 2: Finding values of \(x\)

Now, apply the zero property of multiplication. This means if a product is zero, then at least one of the factors must be zero. Therefore, solve the two equations \(x-5=0\) and \(x+3=0\). This gives two possible solutions for \(x\): \(x=5\) or \(x=-3\)

Step 3: Checking the Solutions

These solutions need to be checked. Substitute \(x=5\) into the original equation: \(5^{2} - 2*5 - 15 = 0\), which is true, so \(x = 5\) is a valid solution. Repeat this process for \(x=-3\): \((-3)^{2} - 2*(-3) - 15 = 0\), which is also true, indicating that \(x = -3\) is a valid solution as well.

Key Concepts

Factoring Quadratic Equations

Factoring quadratic equations is a technique used to simplify solving equations of the form \(ax^{2}+bx+c=0\). It is a crucial skill in algebra and serves as a foundational building block for understanding higher-level math concepts. To factor a quadratic equation means to break it down into simpler binomial factors that, when multiplied together, give you the original quadratic expression.

Let's take the quadratic equation from our exercise \(x^{2}-2x-15=0\). To factor it, we look for two numbers that both add up to the coefficient of the \(x\) term, which is -2, and multiply to give the constant term, which is -15. After a bit of trial and error, we discover that -5 and 3 are the numbers that satisfy both conditions. Therefore, we can express the factored form as \( (x-5)(x+3)=0\). This step is pivotal because it sets the stage for applying the zero product property to find the solutions of the equation.

Zero Product Property

The zero product property is an essential rule in algebra which states that if the product of two factors is zero, then at least one of the factors must be zero. It's a simple yet powerful tool that allows us to solve quadratic equations once we've factored them correctly.

With our factored equation \( (x-5)(x+3)=0\), we apply the zero product property. We set each factor equal to zero and solve for \(x\): \(x-5=0\) gives us \(x=5\), and \(x+3=0\) gives us \(x=-3\). These are the two solutions to our original equation. Whenever you have a factored quadratic equation, remember that the zero product property is your next step to finding the roots of the equation.

Graphing Utility for Intercepts

Using a graphing utility can be highly beneficial when solving quadratic equations. It is particularly useful for visualizing the relationship between the equation and its roots or \(x\)-intercepts. An \(x\)-intercept is a point at which the graph of the equation crosses the \(x\)-axis, which happens when the output value, or \(y\), is zero.

Let's consider our solved equation \(x^{2}-2x-15=0\). If we were to graph this quadratic function, the points where the curve intersects the \(x\)-axis would correspond to the solutions we found by factoring: \(x=5\) and \(x=-3\). These intercepts are the graphical representation of our solutions and serve as a visual confirmation. By identifying \(x\)-intercepts using a graphing utility, students gain a deeper understanding of quadratic functions and appreciate another method to verify their solutions.