Problem 4 - Intermediate Algebra

Question

Solve each of the quadratic equations by factoring and applying the property, \(a b=0\) if and only if \(a=0\) or \(b=0\). If necessary, return to Chapter 3 and review the factoring techniques presented there. $$ x^{2}=15 x $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = 15\).

Step 1: Bring the Equation to Standard Form

The given equation is \(x^2 = 15x\). To start factoring, we need to bring all terms to one side of the equation. Subtract \(15x\) from both sides to get \(x^2 - 15x = 0\).

Step 2: Factor the Quadratic Expression

The equation \(x^2 - 15x = 0\) can be factored by taking \(x\) as a common factor. This gives us \(x(x - 15) = 0\).

Step 3: Apply the Zero Product Property

According to the zero product property, if \(a \cdot b = 0\), then either \(a = 0\) or \(b = 0\). Here, apply this property to \(x(x - 15) = 0\), giving us two equations: \(x = 0\) or \(x - 15 = 0\).

Step 4: Solve Each Equation for x

Solve each of the equations obtained from the previous step. From \(x = 0\), we already have a solution. From \(x - 15 = 0\), add \(15\) to both sides to get \(x = 15\).

Step 5: Verify Solutions

Verify the solutions by substituting back into the original equation. For \(x = 0\): \(0^2 = 0\) matches \(15 \times 0 = 0\). For \(x = 15\): \(15^2 = 225\) matches \(15 \times 15 = 225\). Both solutions satisfy the original equation.

Key Concepts

Factoring

In mathematics, factoring is a method used to break down expressions or numbers into their components, which are called factors. In the context of quadratic equations, factoring is a crucial technique to simplify and solve these expressions.

Quadratic equations are polynomial equations of degree 2, typically written in the form of \( ax^2 + bx + c = 0 \). Factoring helps to simplify this equation by expressing it as a product of two binomials or a monomial and a binomial. Here's a simple breakdown of the factoring process:

• First, rewrite the quadratic equation in standard form \( ax^2 + bx + c = 0 \) if it's not already.
• Identify a common factor, if any, in the terms of the quadratic expression.
• Factor the quadratic equation by expressing it as a product of its factors.

In the example \( x^2 = 15x \), after bringing it to standard form \( x^2 - 15x = 0 \), we find \( x \) as a common factor. Factoring the equation then gives \( x(x - 15) = 0 \). This process transforms the quadratic equation into simpler terms that are easier to handle while solving.

Zero Product Property

The zero product property is a fundamental principle in algebra that states if a product of two or more terms is zero, then at least one of the terms must be zero. This rule is particularly useful when solving quadratic equations that have been factored.

By applying the zero product property, we can break down a complex problem into simpler ones. If \( a \cdot b = 0 \), then either \( a = 0 \) or \( b = 0 \) or both. This logic helps in finding potential solutions for quadratic equations.
In our example, once the equation \( x^2 - 15x = 0 \) is factored into \( x(x - 15) = 0 \), the zero product property tells us that:

• \( x = 0 \)
• \( x - 15 = 0 \)

From these expressions, we can quickly identify that the possible solutions for \( x \) are 0 and 15. The zero product property drastically simplifies the process of solving quadratic equations, turning a single quadratic equation into multiple much simpler linear equations.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of the variable that make the equation true. These equations typically have two solutions, due to their squared variable term, which is reflected as the degree of the polynomial.

Here's a general approach to solving quadratic equations by factoring:

• Ensure the equation is in standard form: \( ax^2 + bx + c = 0 \).
• Factor the quadratic expression into simpler binomials or a monomial and a binomial.
• Use the zero product property to set each factor equal to zero, resulting in one or more simpler equations.
• Solve these equations to find the values of the unknown variable.
• Verify your solutions by substituting them back into the original equation to ensure they satisfy it.

In our specific example, we began with the equation \( x^2 = 15x \), which was rearranged and factored to \( x(x - 15) = 0 \). Applying the zero product property gave us \( x = 0 \) and \( x = 15 \) as solutions. Checking each solution by substituting back into the original equation confirmed they were correct. Successfully solving quadratic equations requires careful attention to each step, particularly during factoring and applying the zero product property.