Problem 18 - Elementary Algebra

Question

For the following problems, solve the equations, if possible. $$ (3 x+2)(x-1)=0 $$

Step-by-Step Solution

Answer

Answer: The solutions to the equation are x=-2/3 and x=1.

Step 1: Apply the zero-product property

Since \((3x+2)(x-1)=0\), either \((3x+2)=0\) or \((x-1)=0\).

Step 2: Solve the first equation

To find the value of \(x\) that satisfies \((3x+2)=0\), subtract 2 from both sides and then divide by 3: $$ 3x+2=0 \\ 3x=-2 \\ x = -\frac{2}{3} $$

Step 3: Solve the second equation

To find the value of \(x\) that satisfies \((x-1)=0\), add 1 to both sides: $$ x-1=0 \\ x=1 $$

Step 4: Write the solution

The solutions to the equation \((3x+2)(x-1)=0\) are \(x=-\frac{2}{3}\) and \(x=1\).

Key Concepts

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra that involves finding the value of the variable that makes the equation true. The key to solving these equations is to isolate the variable on one side, typically by using inverse operations. For instance, if you have an equation like \(3x + 2 = 0\), you would aim to get \(x\) by itself on one side of the equation. Here's how you can do it:

• First, use subtraction to eliminate any constants from the side with the variable. In this case, subtract 2 from both sides to get \(3x = -2\).
• Next, divide by the coefficient of the variable (the number in front of \(x\)). Here, you would divide by 3 to solve for \(x\), resulting in \(x = -\frac{2}{3}\).

Breaking down each step methodically ensures clarity and accuracy. Linear equations can have one solution, no solution, or infinitely many solutions, but for simple cases, like the one above, there tends to be one unique solution.

Factoring

Factoring is a powerful tool used to simplify or solve algebraic expressions and equations. It involves breaking down a complex expression into simpler expressions that, when multiplied together, give the original expression.In the context of the equation \((3x+2)(x-1)=0\), the expression is already factored. This means it is represented as the product of two binomials. Factoring is often used to solve quadratic equations by setting each factor to zero, using the Zero-Product Property.Here's why factoring is handy:

• It helps in simplifying expressions by finding common factors.
• It can transform a problem into a series of smaller problems that are easier to manage.
• Factoring can also help in recognizing and working with patterns and properties of numbers, such as finding roots or intercepts in coordinate geometry.

Understanding how and when to factor plays a crucial role in efficiently solving polynomial equations.

Quadratic Equations

Quadratic equations are a type of polynomial equation of degree two and can be written in the standard form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). These equations can have different types of solutions, depending on the discriminant (the part under the square root in the quadratic formula) and the nature of the roots.The problem you encountered—\((3x + 2)(x - 1) = 0\)—is a factored form of a quadratic equation. When a quadratic equation is factored, solving it becomes more straightforward due to the Zero-Product Property, which states that if a product is zero, then at least one of the factors must be zero. Thus, set each factor to zero and solve for the variable.Characteristics of quadratic equations include:

• They are often solved by factoring, completing the square, or using the quadratic formula.
• They graph as parabolas when plotted on a coordinate plane.
• Depending on the value of the discriminant \((b^2 - 4ac)\), the equation can have two real solutions, one real solution, or no real solutions.

Understanding quadratic equations is critical, as they frequently appear in various mathematical contexts and real-world applications.