Problem 63 - Algebra 1

Question

Solve the quadratic equation. $$13 x^{2}-26 x=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 0\) and \(x = 2\).

Step 1: Set the Equation to Zero

Given the equation \(13 x^{2}-26 x = 0\), it is already set to zero.

Step 2: Factor the Equation

Factor out the common factor, which in this case is \(13x\). So, the equation becomes : \(13x (x - 2) = 0\).

Step 3: Find the Roots of the Equation

Next, set the factored elements to zero and solve for \(x\). This gives \(x = 0\) and \(x = 2\).

Key Concepts

Factoring quadratic equations

Factoring is a method used to rewrite a quadratic equation as a product of its factors, which can simplify the process of solving it. In our example, the equation is \(13x^2 - 26x = 0\). Factoring involves searching for a common term that can be taken out from each part of the quadratic. Here, the common factor is \(13x\).

This leads us to express the equation as \(13x(x-2) = 0\). When factoring, ensure that what you're multiplying (the factors) can be expanded back to the original equation. This check helps confirm the accuracy of your factoring. Once you have correctly factored the equation, it makes the next steps of finding the roots significantly easier.

Finding roots

Finding roots of a quadratic equation means determining the values of \(x\) that make the equation true. After factoring the equation \(13x(x-2) = 0\), we use the Zero Product Property. This property states that if a product of two factors is zero, at least one of the factors must be zero.

To find the roots:

• Set each factor equal to zero:
• \(13x = 0\)
• \(x - 2 = 0\)

This results in two potential solutions. From \(13x = 0\), we derive \(x = 0\). From \(x - 2 = 0\), we solve and find \(x = 2\). Thus, the roots of the equation are \(x = 0\) and \(x = 2\). These are the points where the quadratic function crosses the x-axis.

Solving quadratic equations

Solving quadratic equations usually means finding an equation's roots or solutions. There are several strategies to solve a quadratic equation, such as factoring, using the quadratic formula, or completing the square. In this exercise, we used the factoring method, which is one of the most direct and simplest methods when applicable.

The goal of solving a quadratic equation is to express it in terms of its roots. After factoring \(13x^2 - 26x = 0\) into \(13x(x-2) = 0\), and finding \(x = 0\) and \(x = 2\), we reached the solution. It is important to choose the method that matches best with the equation given. Some quadratics are easier to solve using the quadratic formula, particularly when factoring is not straightforward. Each method has its utilities, so being familiar with multiple techniques enhances your capability to solve any quadratic equation efficiently.