Problem 63 - Algebra 1: Concepts and Skills

Question

Choose a method and solve the quadratic equation. Explain your choice. $$ x^{2}+5 x-14=0 $$

Step-by-Step Solution

Answer

The solutions to the equation \(x^2 + 5x - 14 = 0\) are \(x = 2\) and \(x = -7\).

Step 1: Set Up the Equation

First we need to rewrite the equation. It is already in the standard form which is \(x^2 + 5x - 14 = 0\).

Step 2: Factor the Quadratic Equation

The factored form of a quadratic equation is \((x - p)(x - q) = 0\), where \(p\) and \(q\) are the solutions of the equation. Let's find two numbers that multiply to -14 (the value of \(c\)) and add up to 5 (the value of \(b\)). After a little trial and error, we find that these two numbers are -2 and 7. Thus we can rewrite the equation as follows: \((x - 2)(x + 7) = 0\).

Step 3: Solve for x

We set each factor equal to zero and solve for \(x\): (1) \(x - 2 = 0\), which solves to \(x = 2\). And (2) \(x + 7 = 0\), which solves to \(x = -7\).

Key Concepts

Factoring Quadratics

Factoring quadratics is a key technique in algebra that helps simplify quadratic equations, such as \(x^2 + 5x - 14 = 0\). A quadratic equation is typically expressed in the standard form as \(ax^2 + bx + c = 0\). To factor it means to express it as the product of two binomial expressions. The main goal is to find two numbers, let's call them \(p\) and \(q\), that multiply to give you the constant term \(c\), and also add up to the linear coefficient \(b\).

• In our example, we look for two numbers that multiply to \(-14\) and add up to \(5\).
• After testing several combinations, we identify \(-2\) and \(7\) as the numbers that fulfill these criteria.

This allows us to effectively rewrite the quadratic as \((x - p)(x + q) = 0\), or \((x - 2)(x + 7) = 0\). Factoring makes solving the equation convenient, as we can then use other algebraic methods like the Zero Product Property to find the solutions.

Solving Equations

Once a quadratic equation is factored, solving it becomes straightforward. The Zero Product Property is our best friend here: if the product of two numbers is zero, then at least one of the numbers must be zero. For the factored equation \((x - 2)(x + 7) = 0\), we set each part of the equation separately to zero:

• \(x - 2 = 0\)
• \(x + 7 = 0\)

By solving these simple linear equations, we discover that:

• Setting \(x - 2 = 0\), we find \(x = 2\).
• Setting \(x + 7 = 0\), we find \(x = -7\).

These solutions, \(x = 2\) and \(x = -7\), represent the x-values where the original quadratic equation equals zero. This method offers a reliable way to find the roots of the equation efficiently.

Algebraic Methods

Algebraic methods, like factoring, are essential tools in solving quadratic equations. These methods provide a structured approach to manipulate and simplify equations to find solutions more easily. While there are various ways to tackle a quadratic equation, each has its pros and cons.

• Factoring is a quick method when the equation is simple and the factors are easy to identify.
• In cases where the quadratic is difficult to factor, other algebraic methods like completing the square or using the quadratic formula may be more appropriate.
• The choice of method depends on the specific equation and sometimes personal preference if the problem can be solved in multiple ways.

Understanding these algebraic techniques enriches problem-solving skills and gives students the flexibility to choose the most efficient method for each situation. This variety in approach is what makes algebra both powerful and versatile in mathematics.