Problem 9 - Elementary Algebra

Question

Use the zero-factor property to solve each equation. $$ (x+5)(3 x-4)=0 $$

Step-by-Step Solution

Answer

Answer: The values of x that satisfy the equation are x = -5, and x = 4/3.

Step 1: Apply the zero-factor property

Set each factor equal to zero: $$ (x+5) = 0 \\ (3x-4) = 0 $$

Step 2: Solve for x in the first equation

Subtract 5 from both sides to isolate x: $$ x + 5 - 5 = 0 - 5 \\ x = -5 $$

Step 3: Solve for x in the second equation

Add 4 to both sides and then divide by 3 to isolate x: $$ 3x - 4 + 4 = 0 + 4 \\ 3x = 4 \\ x = \frac{4}{3} $$

Step 4: Write the final solution

The values of x that satisfy the given equation are: $$ x = -5, \frac{4}{3} $$

Key Concepts

Solving Equations

Solving equations is a core skill in mathematics, invaluable for anyone learning algebra. It often involves finding the values of variables that make an equation true. In our example, the equation \((x + 5)(3x - 4) = 0\), we need to discover the values of \(x\) that satisfy this equation.
This equation is understood as having solutions where either one or both factors result in the equation equalling zero. A systematic approach involves:

• Setting each factor separately equal to zero.
• Solving each resulting simpler equation to find possible values of \(x\).

In this way, solving equations can sometimes involve dividing the solution into smaller, manageable parts to reach an accurate solution.

Factorization

Factorization is the process of breaking down an expression into simpler, multiplied parts called factors. Think of it like reverse multiplication. When you factor an expression, you express it as a product of its factors. For example, the equation \((x + 5)(3x - 4) = 0\) is already factorized.
Factorization makes it easier to use the zero-factor property, which states: if a product of multiple factors equals zero, then at least one of the factors must be zero. This turns a complex equation into simpler equations that are much easier to solve.
In practice:

• Look for common factors.
• Use special formulas like difference of squares or quadratic formulas if applicable.
• Write the expression as a product of its factors.

Understanding factorization is key to simplifying and solving algebraic equations efficiently.

Algebraic Equations

Algebraic equations form the basis of algebra and involve expressions containing variables and constants. In solving the equation \((x + 5)(3x - 4) = 0\), we're dealing with a basic form of algebraic equation.
The steps involve:

• Identifying the structure of the equation.
• Applying mathematical operations systematically.
• Using properties like the zero-factor property to unravel solutions.

Algebraic equations can be linear, quadratic, or more complex. Here, we solved a quadratic equation by leveraging its factorized form.
In general, understanding how equations are structured and the properties that apply to them is crucial for efficiently finding solutions and understanding the fundamental principles of algebra.