Problem 37 - Intermediate Algebra

Question

Simplify and reduce each expression. $$ 5 x^{2}-13 x=0 $$

Step-by-Step Solution

Answer

x = 0 or x = \frac{13}{5}.

Step 1: Identify Common Factors

The given equation is \(5x^2 - 13x = 0\). First, identify the common factor in both terms. We can see that \(x\) is a common factor.

Step 2: Factor Out the Common Factor

Factor \(x\) out of the equation to simplify it. This gives: \(x(5x - 13) = 0\).

Step 3: Apply the Zero Product Property

According to the zero product property, if \(a \cdot b = 0\), then either \(a = 0\) or \(b = 0\). Apply this property to the equation \(x(5x - 13) = 0\).

Step 4: Solve Each Factor for Zero

Solve the equation \(x = 0\). Then, solve the second factor, \(5x - 13 = 0\), which gives \(5x = 13\).

Step 5: Solve for x

To find the value of \(x\), divide both sides of the equation by 5, resulting in \(x = \frac{13}{5}\). Thus, the solutions are \(x = 0\) and \(x = \frac{13}{5}\).

Key Concepts

Zero Product Property

The Zero Product Property is a mathematical principle that says if the product of two or more numbers is zero, then at least one of those numbers must be zero. This property is particularly useful when solving quadratic equations or any polynomials factored into simpler terms.
In the solved exercise with the equation \(x(5x - 13) = 0\), we use the Zero Product Property to set each factor to zero:

• \(x = 0\)
• \(5x - 13 = 0\)

Once we have these two equations, it’s straightforward to find the solutions for \(x\). In this case, these solutions are \(x = 0\) and \(x = \frac{13}{5}\). Using the Zero Product Property helps to simplify solving equations and finding all possible solutions quickly.

Factoring Quadratics

Factoring quadratics involves rewriting a quadratic expression as a product of two binomials. This approach is highly effective in solving quadratic equations. By factoring, you transform an equation into a form where the Zero Product Property can be applied easily.
In the example of the equation \(5x^2 - 13x = 0\), factoring begins with identifying the common factors. We noticed that \(x\) is common, so we factor it out to get \(x(5x - 13) = 0\).
Factoring, in this case, requires recognizing patterns or using special techniques such as the greatest common factor to break down the quadratic expression into simpler multiplicative components. This method not only reveals solutions efficiently but also deepens our understanding of the equation's structure.

Common Factor Identification

Identifying common factors is crucial when working to simplify expressions or solve equations. A common factor is a number or variable that divides exactly into each term of the expression, leading to simplification.
In the equation given: \(5x^2 - 13x = 0\), the variable \(x\) is present in each term. Recognizing this allows us to factor it out of the equation.

• This process reduces the expression to \(x(5x - 13) = 0\), simplifying the equation.
• Finding common factors reduces complexity and makes it easier to continue with the problem-solving process.

Identifying and factoring out common elements is an essential skill in mathematics that aids in solving various types of equations efficiently.