Problem 22 - Algebra for College Students

Question

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$a(a-1)=2$$

Step-by-Step Solution

Answer

The solutions are \( a = 2 \) and \( a = -1 \).

Step 1: Rewrite the Equation

Start by rewriting the given equation, \( a(a-1) = 2 \), as a standard quadratic expression by expanding and setting it to zero: \( a^2 - a = 2 \).

Step 2: Set the Equation to Zero

Move all terms to one side to form a quadratic equation equal to zero: \( a^2 - a - 2 = 0 \).

Step 3: Identify a Factoring Approach

Look for two numbers that multiply to \(-2\) and add to \(-1\). These numbers are \(-2\) and \(1\).

Step 4: Factor the Quadratic Equation

Factor the quadratic expression \( a^2 - a - 2 \) using the numbers found. It can be rewritten as \( (a - 2)(a + 1) = 0 \).

Step 5: Apply the Zero Product Property

Set each factor equal to zero and solve for \(a\):\( a - 2 = 0 \) or \( a + 1 = 0 \).

Step 6: Solve for Variable a

Solve the equations from the previous step: \( a = 2 \) or \( a = -1 \).

Key Concepts

Factoring Techniques

Factoring is a powerful technique for simplifying expressions, especially in quadratic equations. To factor a quadratic expression, you need to express it as a product of two binomials. When you factor, you look for two numbers that multiply to the constant term of the quadratic equation and add up to the coefficient of the linear term. For example, consider the equation from the exercise:

• We have the quadratic expression in the form of \(a^2 - a - 2\).
• The task involves finding two numbers that multiply to \(-2\) and add up to \(-1\).
• These numbers are \(-2\) and \(1\), providing the factors \((a - 2)(a + 1)\).

This process helps convert the quadratic into a form that’s easier to solve using other algebraic methods.

Zero Product Property

The Zero Product Property is a fundamental principle used to solve equations that have been factored. It states that if the product of two numbers is zero, then at least one of the numbers must be zero. Applying it to our factored equation, \((a - 2)(a + 1) = 0\):

• According to the property, either \(a - 2 = 0\) or \(a + 1 = 0\).
• This allows us to set each factor separately equal to zero as distinct equations to solve for \(a\).

This property is crucial because it simplifies the solving process, taking advantage of the structure created by factoring.

Solving Equations

Solving equations involves finding the values that make the equation true. After factoring the quadratic expression and applying the zero product property, you are left with simpler linear equations that can be solved directly. Here's how you solve the factored equation \((a - 2)(a + 1) = 0\):

• For \(a - 2 = 0\), add \(2\) to both sides to get \(a = 2\).
• For \(a + 1 = 0\), subtract \(1\) from both sides to get \(a = -1\).

These solutions, \(a = 2\) and \(a = -1\), are the values that satisfy the original quadratic equation.

Quadratic Expressions

A quadratic expression is any equation that can be written in the standard form \(ax^2 + bx + c = 0\). These expressions often arise in various mathematical contexts, and solving them is a key skill. In terms of our exercise, we began with a factorable form, \(a(a-1) = 2\), which translates into the standard form:

• Expanding \(a(a-1)\) gives it a quadratic structure: \(a^2 - a\).
• Then, rearranging gives \(a^2 - a - 2 = 0\).

Understanding how to manipulate and solve quadratic expressions lays the groundwork for tackling more complex algebraic problems.