Problem 8 - Elementary Algebra

Question

For the following problems, solve the equations. $$ (5 a-2)(3 a-10)=0 $$

Step-by-Step Solution

Answer

Answer: The solutions for the given equation are a = 2/5 and a = 10/3.

Step 1: Analyze the equation

First, look at the equation given: $$ (5a - 2)(3a - 10) = 0 $$ Notice that it is in the factored form, which allows us to apply the zero-product property. The zero-product property states that if the product of two factors is zero, then one or both of the factors must be zero.

Step 2: Apply the zero-product property

Now, apply the zero-product property. We will set each factor, \((5a - 2)\) and \((3a - 10)\), equal to zero: $$ 5a - 2 = 0 \\ 3a - 10 = 0 $$

Step 3: Solve for the variable 'a'

Next, let's solve each equation for the variable 'a': For the first equation, $$ 5a - 2 = 0 \\ 5a = 2 \\ a = \frac{2}{5} $$ For the second equation, $$ 3a - 10 = 0 \\ 3a = 10 \\ a = \frac{10}{3} $$

Step 4: Write down the solutions

Finally, write down the solutions for the variable 'a': $$ a = \frac{2}{5}, \frac{10}{3} $$ These are the two possible solutions to the given equation.

Key Concepts

Factored Form Equations

Factored form equations are a special way of expressing equations, particularly quadratic ones. This form is popular and quite useful because it breaks down equations into simpler, multiplied factors. For example, consider the equation \((5a - 2)(3a - 10) = 0\).
In this equation, the product of two expressions, \(5a - 2\) and \(3a - 10\) is equal to zero.
The beauty of factored form equations lies in their simplicity when it comes to finding the roots or solutions of the equation. Instead of expanding and dealing with a possibly messy polynomial, you work directly with each factor.

• The equations are already broken down, which makes them much easier to solve.
• Each bracketed expression, or factor, represents part of the solution.

Understanding the factored form means you recognize an equation as a product of individual terms that set up perfectly for using the zero-product property.

Solving Quadratic Equations

When you're solving quadratic equations like\((5a - 2)(3a - 10) = 0\), you're finding the values of \(a\) that make the equation true.
The structure of the equation, already in factored form, leads to an efficient solution path. The zero-product property is a fundamental concept here. This rule tells us that for a product to equal zero, at least one of the factors must be zero.So, to solve the equation:

• Set each factor equal to zero:
• \(5a - 2 = 0\) and \(3a - 10 = 0\).

These steps turn one equation into two simpler linear equations. The solutions of these linear equations will give all possible values of \(a\) that satisfy the original quadratic equation.

Algebraic Solutions

Finding algebraic solutions requires us to work through the steps described in the previous sections, using basic algebra skills.
For example, starting with \(5a - 2 = 0\), you can solve for \(a\) by isolating the variable. Move constants to one side and divide by the coefficient of \(a\) to get \(a = \frac{2}{5}\).
The second factor, \(3a - 10 = 0\), is solved similarly, resulting in \(a = \frac{10}{3}\).
By following these straightforward algebraic manipulations, we derive the two solutions \(a = \frac{2}{5}\) and \(a = \frac{10}{3}\).Breaking the equation down and solving through each factor using algebra highlights how logical steps lead to clear answers.

• Identify the factor.
• Isolate the variable.
• Perform arithmetic operations.

These solutions are valuable examples of how algebraic methods can simplify what might appear to be complex problems. By adhering to these steps, the process becomes an organized way to tackle quadratic equations efficiently.