Problem 15 - Contemporary Precalculus

Question

Solve the equation by factoring. $$x^{2}-5 x=14$$

Step-by-Step Solution

Answer

Question: Solve the quadratic equation $$x^2 - 5x = 14$$. Answer: The solutions to the given quadratic equation are $$x = 7$$ and $$x = -2$$.

Step 1: Rewrite the equation in the standard form

Subtract 14 from both sides of the equation to rewrite it in the standard form: $$x^2 - 5x - 14 = 0$$

Step 2: Factor the quadratic expression

We need to find two numbers that multiply to -14 and add to -5. These numbers are -7 and 2. Use these numbers to factor the quadratic expression: $$(x-7)(x+2) = 0$$

Step 3: Use the Zero Product Property to find the solutions

According to the Zero Product Property, if the product of two factors is zero, then at least one of the factors is zero. So, we have: 1. $$x - 7 = 0$$ 2. $$x + 2 = 0$$ Solve these two simple equations to find the values of x that make the original equation true.

Step 4: Solve for x

Solve both simple equations to find the values of x: 1. $$x - 7 = 0 \Rightarrow x=7$$ 2. $$x + 2 = 0 \Rightarrow x=-2$$ The solutions to the given quadratic equation are: $$x = 7$$ and $$x = -2$$

Key Concepts

Factoring Quadratics

Factoring quadratics is a method to solve quadratic equations by expressing the quadratic in a product form. A quadratic equation is typically given in the form \(ax^2 + bx + c = 0\). In this form, our task is to find two numbers that multiply to the constant term \(c\) and add up to the linear coefficient \(b\). These numbers help transform the equation into a factorable format.

In the exercise, the equation was \(x^2 - 5x = 14\). We first brought it to standard form by subtracting 14, resulting in \(x^2 - 5x - 14 = 0\). Here, we needed to find two numbers whose product is \(-14\) (the constant term) and sum is \(-5\) (the linear coefficient).

• The two numbers found were \(-7\) and \(2\).
• These allow the expression to be factored into \((x - 7)(x + 2)\).

This step simplifies solving the quadratic equation by breaking down a more complex expression into manageable parts.

Zero Product Property

The Zero Product Property is a useful mathematical rule that helps us find solutions to equations once they have been factored.

This property states that if the product of two numbers is zero, at least one of the numbers must be zero. This means if you have factored a quadratic equation into two binomials, you can set each binomial equal to zero to find the solutions of the equation.

Using the exercise's factorization \((x - 7)(x + 2) = 0\):

• First equation: \(x - 7 = 0\), which implies \(x = 7\).
• Second equation: \(x + 2 = 0\), which simplifies to \(x = -2\).

By applying the Zero Product Property here, we convert a single quadratic equation into two linear equations that are easily solvable. This concept is fundamental for dealing with factorable quadratics.

Solving Equations

Solving equations involves finding the values of the variable that make the equation true. In our example of a quadratic equation \(x^2 - 5x - 14 = 0\), we applied factoring followed by the Zero Product Property to find these solutions.

Once the quadratic is expressed in its factored form \((x - 7)(x + 2) = 0\), solving the equation becomes straightforward:

• For \(x - 7 = 0\), solving gives \(x = 7\).
• For \(x + 2 = 0\), solving gives \(x = -2\).

Hence, the solutions to the equation are \(x = 7\) and \(x = -2\).

Solving such equations is an essential part of algebra, especially as it lays the foundation for solving more complex quadratic equations that appear in higher-level mathematics. Recognizing the right methods to employ, such as factoring and applying the Zero Product Property, facilitates a clear path toward finding accurate solutions.