Problem 59 - Elementary Algebra

Question

For the following problems, use the zero-factor property to solve the equations. $$ (3 x-1)(3 x-1)=0 $$

Step-by-Step Solution

Answer

Answer: x = 1/3

Step 1: Understand the zero-factor property

The zero-factor property states that if the product of two or more factors is zero, then at least one of those factors must be zero.

Step 2: Identify the factors

In the given equation, we have the product of two factors: \((3x-1)\) and \((3x-1)\). We can rewrite the equation as: $$ (3x-1)(3x-1) = 0 $$

Step 3: Apply the zero-factor property to solve the equation

Now, we can set each factor equal to zero: $$ 3x-1 = 0 $$

Step 4: Solve for x

Adding 1 to both sides of the equation, we get: $$ 3x = 1 $$ Now, we can divide both sides by 3 to isolate x: $$ x=\frac{1}{3} $$ Since both factors were identical, we only need to solve for x once. The solution for the given equation is \(x=\frac{1}{3}\).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra, making appearances in various real-world applications. A quadratic equation is any equation that can be rearranged into the standard form:\[ ax^2 + bx + c = 0 \]where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable or unknown. Quadratic equations are defined by the presence of the \(x^2\) term, which gives the equation its quadratic nature.When solving quadratic equations, there are several methods you can use:

• Factoring
• Using the quadratic formula
• Completing the square

The method you choose often depends on the specific equation you are dealing with. In our current example, factoring is the most straightforward approach since the equation is already presented in factored form.Quadratic equations may have two solutions, one solution (where the solutions are repeated), or no real solutions. In this exercise, we found that the solution is repeated, indicating that the parabola of the quadratic equation touches the x-axis at one point.

Solving Equations

Solving equations is the process of finding the value of the variable that makes the equation true. The challenge is to manipulate the equation till we isolate the variable. For our specific equation:\[ (3x-1)(3x-1) = 0 \]We start by acknowledging that the expressions in parentheses are identical. This makes the equation simplified as all we need to do is solve for \(x\) once, stemming from the property that if the product of identical terms is zero, then one or both of those terms must be zero.The procedure involves:

• Setting the equation to zero based on the zero-factor property.
• Solving the resulting simple linear equations independent of each factor.
• Substituting values back to verify solutions if necessary.

In our example, by solving \(3x-1 = 0\), we arrive at the value of \(x = \frac{1}{3}\). This solution satisfies the original equation.

Factoring in Algebra

Factoring in algebra involves expressing a polynomial as the product of its factors. Factors are simpler polynomials or numbers that can be multiplied to give the original polynomial. This technique is particularly useful for solving equations, especially quadratics like in our exercise.To factor, we look for common elements, patterns, or special forms within the polynomial. The zero-factor property plays a crucial role here: it tells us that if a product of factors equals zero, then at least one of the factors must be zero. In the equation \((3x-1)(3x-1) = 0\), the polynomial is already factored. We simply set each factor equal to zero. This reduces the steps to solving simple linear equations, making the process efficient and straightforward.Factoring requires practice and familiarity with different types of polynomials:

• Perfect square trinomials
• Difference of squares
• Common factors

Mastering these techniques can vastly simplify the process of solving complex algebraic equations.