Problem 3 - Basic Technical Mathematics with Calculus

Question

Find the roots of the given equations by inspection. $$(x+2)\left(x^{2}-9\right)=0$$

Step-by-Step Solution

Answer

The roots are \(x = -2\), \(x = 3\), and \(x = -3\).

Step 1: Analyze the Structure of the Equation

The given equation is \((x+2)(x^2-9)=0\). This is a product of two separate expressions: \((x+2)\) and \((x^2-9)\). According to the zero-product property, if the product of two factors equals zero, at least one of the factors must be zero.

Step 2: Solve the First Factor

Set the first factor equal to zero: \(x+2=0\). Solve for \(x\) by subtracting 2 from both sides, giving \(x = -2\).

Step 3: Solve the Second Factor

Set the second factor equal to zero: \(x^2-9=0\). Notice that this is a difference of squares which can be factored as \((x-3)(x+3)=0\).

Step 4: Solve the Factored Form of the Second Factor

Set each factor of the equation obtained from the difference of squares to zero. First, \(x-3=0\): solve to find \(x = 3\). Then, \(x+3=0\): solve to find \(x = -3\).

Step 5: Compile All Roots

Collect the roots obtained from all the factors: \(x = -2\), \(x = 3\), and \(x = -3\). These are the solutions to the equation \((x+2)(x^2-9)=0\).

Key Concepts

Zero-Product Property

The zero-product property is a crucial concept when solving quadratic equations, especially in factored form like \((x+2)(x^2-9)=0\). This property states that if the product of two or more factors equals zero, at least one of the factors must be zero.

• If \((a\cdot b = 0)\), then either \(a = 0\) or \(b = 0\) (or both).
• In our problem, the two factors are \((x+2)\) and \((x^2-9)\).

By setting each factor equal to zero, we break down the problem into smaller, more manageable parts. Solving each equation:

• For \(x+2=0\), subtract 2 from both sides to find \(x = -2\).
• The second factor \(x^2-9=0\) involves factoring further, which leads us to our next step.

Factoring Quadratics

Factoring quadratics involves rewriting a quadratic expression as a product of simpler expressions.

• When given a quadratic equation like \(x^2 - 9 = 0\), factoring helps us express it as a product of binomials.
• Such expressions are easier to solve using the zero-product property.

In our exercise, the expression \(x^2 - 9\) is used. This is a special type of quadratic known as a difference of squares, which we will explore next. Identifying patterns makes factoring quicker and more intuitive:

• For a standard quadratic \(ax^2 + bx + c = 0\), seek two numbers that multiply to \(ac\) and add to \(b\).
• In our example, the expression factors into \((x-3)(x+3)\), making the solution straightforward.

Difference of Squares

The difference of squares is a specific method of factoring quadratics and an essential skill for solving certain types of equations. It applies when you have expressions of the form \(a^2 - b^2\). This can be factored as:

• \(a^2 - b^2 = (a-b)(a+b)\)

In our exercise, \(x^2 - 9\) fits this category:

• \(x^2\) is \(x\) squared and \(9\) is \(3\) squared.
• Therefore, \(x^2 - 9\) becomes \((x-3)(x+3)\).

This step simplifies the equation significantly. When each factor is set to zero, solving becomes a matter of simple algebra:

• Set \(x-3=0\) and solve to get \(x=3\).
• Set \(x+3=0\), resulting in \(x=-3\).

Recognizing and applying the difference of squares keeps this process efficient, allowing you to find solutions quickly.