Problem 9 - Elementary Algebra

Question

Solve \((9 x+2)(7 x-3)=0\) using this mental method.

Step-by-Step Solution

Answer

Answer: The two solutions for the equation are \(x = \frac{-2}{9}\) and \(x = \frac{3}{7}\).

Step 1: Solve for the first factor

To solve the equation \((9x+2)=0\), isolate \(x\) by subtracting 2 from both sides, and then divide both sides by 9: \[x= \frac{-2}{9}\]

Step 2: Solve for the second factor

To solve the equation \((7x-3)=0\), isolate \(x\) by adding 3 to both sides, and then divide both sides by 7: \[x= \frac{3}{7}\] Thus, the two solutions for the given equation \((9x+2)(7x-3) = 0\) are \(x = \frac{-2}{9}\) and \(x = \frac{3}{7}\).

Key Concepts

Factoring

Factoring is an essential method in algebra used to simplify mathematical expressions or solve equations. It involves expressing a polynomial as a product of its factors. For quadratic equations, factoring can help break down a complex problem into simpler, more manageable linear equations. In many cases, identifying common factors, recognizing difference of squares, or applying special patterns like the trinomial can lead to successful factoring.

• Breaking down: Express the equation as a product of its factors, like (9x+2)(7x-3) in the given exercise.
• Pattern Recognition: Recognize patterns and use them to simplify the factoring process.

The main goal of factoring in problems like these is to express the equation as the product of two or more linear expressions. This makes it easier to apply the Zero Product Property and solve for the variable.

Zero Product Property

The Zero Product Property is an important concept that helps solve equations quickly and efficiently. This property states that if a product of two factors equals zero, then at least one of the factors must be zero. It's a powerful tool used mainly after factoring a quadratic equation. Here is how it works:

• After factoring, set each factor equal to zero.
• Each factor represents a potential solution.

In the exercise example, the equation (9x+2)(7x-3) equals zero. By setting each factor — 9x+2 and 7x-3 — equal to zero, we can solve for the variable (x). Each of these calculations provides a possible solution to the quadratic equation and confirms that at least one of the factor expressions results in zero.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. Once a quadratic equation is factored and the Zero Product Property is applied, you're only left with simple linear equations to solve.

• Isolate the variable: Move terms around to one side of the equation to solve for the variable x.
• Perform arithmetic operations: Use addition, subtraction, multiplication, or division to simplify the equation.

For the two linear factors in the exercise, we solve 9x+2 = 0 and 7x-3 = 0 by isolating x. This gives us x = \( \frac{-2}{9} \) and x = \( \frac{3}{7} \) respectively, which are the solutions to the original quadratic equation. By learning to efficiently solve linear equations, you unravel the solution to more complex problems.