Problem 14 - Algebra A Combined Function

Question

Solve each equation. $$ (9 x+1)(4 x-3)=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = -\frac{1}{9}\) and \(x = \frac{3}{4}\).

Step 1: Identify the Zero Product Property

When the product of two terms is zero, at least one of the terms must be zero. Thus, if \((9x + 1)(4x - 3) = 0\), then \((9x + 1) = 0\) or \((4x - 3) = 0\).

Step 2: Solve the First Equation

Solve \(9x + 1 = 0\) by subtracting 1 from both sides to get \(9x = -1\). Then divide both sides by 9 to find \(x = -\frac{1}{9}\).

Step 3: Solve the Second Equation

Solve \(4x - 3 = 0\) by adding 3 to both sides to get \(4x = 3\). Then divide both sides by 4 to find \(x = \frac{3}{4}\).

Step 4: Conclude the Solution

The solutions to the equation \((9x + 1)(4x - 3) = 0\) are \(x = -\frac{1}{9}\) and \(x = \frac{3}{4}\).

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra, typically taking the form \(ax^2 + bx + c = 0\). They can appear complex at first, but with a few strategies, they become more manageable. In the given exercise, the equation is transformed into a factored format: \((9x + 1)(4x - 3) = 0\). This shows a special characteristic of quadratic equations: they can often be expressed as the product of two binomials. This structure is key, making it easier to solve the equation by applying the zero product property. It is essential to recognize that when an equation appears in this factored form, it often simplifies the task of finding the variable's values that satisfy the equation. As you work through more quadratic equations, the ability to factor them quickly and accurately becomes an indispensable skill.

Algebraic Solutions

Solving equations algebraically involves a series of logical steps to isolate the variable. For this exercise, after establishing the zero product property, we solve two separate linear equations: \(9x + 1 = 0\) and \(4x - 3 = 0\). This approach breaks down complex problems into simpler, more manageable parts.

• To solve \(9x + 1 = 0\), subtract 1 from both sides, yielding \(9x = -1\).
• Next, divide each side by 9 to isolate x, giving \(x = -\frac{1}{9}\).
• For \(4x - 3 = 0\), add 3 to both sides, resulting in \(4x = 3\).
• Finally, divide both sides by 4 to find \(x = \frac{3}{4}\).

This step-by-step method ensures that each solution is logically and systematically derived, helping you understand the underlying problem-solving techniques.

Factoring

Factoring is a crucial skill in solving quadratic equations. It involves breaking down an expression into simpler terms, or factors, whose product is the original expression. In the given exercise, recognizing that the expression \((9x + 1)(4x - 3)\) represents the product of two factors allows the application of the zero product property. Factoring simplifies equations by showing how different terms contribute to the overall equation.
Learning to factor efficiently involves identifying common patterns or techniques:

• Look for common factors in all terms.
• Apply the difference of squares if applicable.
• Use the quadratic formula if factoring seems complex.

It’s important to practice factoring in different scenarios to build confidence. With time, recognizing the factorable parts of an equation will speed up the problem-solving process.