Solving 27x² + 3 = 18x By Factoring A Step-by-Step Guide
In this comprehensive guide, we will delve into the process of solving the quadratic equation 27x² + 3 = 18x by factoring. Factoring is a fundamental technique in algebra, allowing us to break down complex expressions into simpler components, making it easier to find solutions. This method relies on the principle that if the product of two or more factors is zero, then at least one of the factors must be zero. By applying this principle, we can transform a quadratic equation into a set of linear equations, which are much simpler to solve. This article provides a detailed, step-by-step approach to solving this equation, suitable for students, educators, and anyone interested in enhancing their algebraic skills.
1. Understanding Quadratic Equations
To effectively tackle the equation 27x² + 3 = 18x, it's essential to grasp the fundamentals of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (in this case, 'x') is 2. The standard form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. Understanding this standard form is crucial because it allows us to identify the coefficients and apply various methods to solve the equation. Methods such as factoring, completing the square, and using the quadratic formula are commonly employed, each with its own advantages depending on the specific equation. In our case, we will focus on the factoring method, which is particularly effective when the quadratic expression can be easily factored into simpler terms. Mastering the concept of quadratic equations and their standard form is the first step towards solving them efficiently.
2. Rearranging the Equation into Standard Form
The initial form of our equation, 27x² + 3 = 18x, is not in the standard quadratic form ax² + bx + c = 0. To proceed with factoring, we must first rearrange the equation to fit this standard form. This involves moving all terms to one side of the equation, leaving zero on the other side. The key is to maintain the balance of the equation by performing the same operations on both sides. In this case, we need to subtract 18x from both sides of the equation. This step is crucial because it allows us to clearly identify the coefficients 'a', 'b', and 'c', which are necessary for factoring and other solution methods. By rearranging the equation, we set the stage for applying factoring techniques and ultimately finding the values of 'x' that satisfy the equation. This transformation is a fundamental step in solving any quadratic equation.
Subtracting 18x from both sides gives us:
27x² - 18x + 3 = 0
3. Simplifying the Equation by Factoring out the Greatest Common Factor (GCF)
Before diving into factoring the quadratic expression, it's always a good practice to look for any common factors among the coefficients. This simplification can make the factoring process much easier. In our equation, 27x² - 18x + 3 = 0, we can observe that all three coefficients (27, -18, and 3) are divisible by 3. Factoring out the greatest common factor (GCF), which in this case is 3, not only simplifies the equation but also reduces the complexity of the numbers we are dealing with. This step is a prime example of how a little simplification can lead to significant ease in solving mathematical problems. By dividing each term by the GCF, we obtain a simpler quadratic equation that is much more manageable to factor. This technique is a valuable tool in solving not just quadratic equations but various algebraic expressions as well.
Dividing the entire equation by 3, we get:
9x² - 6x + 1 = 0
4. Factoring the Quadratic Expression
Now that we have simplified the equation to 9x² - 6x + 1 = 0, we can proceed with factoring the quadratic expression. Factoring involves breaking down the quadratic expression into two binomials, such that when multiplied together, they yield the original quadratic expression. This step often requires some trial and error, but with practice, it becomes more intuitive. The key is to find two numbers that multiply to give the constant term (1 in this case) and add up to the coefficient of the linear term (-6 in this case). In this particular equation, the quadratic expression is a perfect square trinomial, which means it can be factored into the square of a binomial. Recognizing such patterns can greatly simplify the factoring process. Mastering factoring techniques is essential for solving quadratic equations and is a fundamental skill in algebra.
The expression 9x² - 6x + 1 is a perfect square trinomial. It can be factored as:
(3x - 1)(3x - 1) = 0
This can also be written as:
(3x - 1)² = 0
5. Solving for x
With the quadratic equation factored as (3x - 1)² = 0, we can now solve for 'x'. The principle we use here is that if the square of a quantity is zero, then the quantity itself must be zero. This allows us to set the binomial factor equal to zero and solve the resulting linear equation. This step is straightforward and involves basic algebraic manipulation. By isolating 'x', we find the value(s) that satisfy the original quadratic equation. In this case, since the factored form is a perfect square, we expect to find one repeated root. Understanding how to extract solutions from factored equations is a critical skill in algebra and is applicable to a wide range of mathematical problems.
Setting the factor equal to zero:
3x - 1 = 0
Adding 1 to both sides:
3x = 1
Dividing both sides by 3:
x = 1/3
6. Final Answer and Interpretation
Therefore, the solution to the quadratic equation 27x² + 3 = 18x is x = 1/3. Since the factored form of the equation resulted in a repeated factor, we have one real solution, also known as a repeated root. This means that the parabola represented by the quadratic equation touches the x-axis at only one point, x = 1/3. Understanding the nature of the solutions, whether they are real, repeated, or complex, is crucial in interpreting the behavior of the quadratic function. This final step consolidates our understanding of the entire process, from rearranging the equation to factoring and finally solving for 'x'. This comprehensive approach ensures a solid grasp of solving quadratic equations by factoring.
So, the solution is:
x = 1/3
Summary
In summary, solving the quadratic equation 27x² + 3 = 18x by factoring involves several key steps: rearranging the equation into standard form, simplifying by factoring out the GCF, factoring the quadratic expression, solving for 'x', and interpreting the solution. This method provides a clear and systematic approach to solving quadratic equations, emphasizing the importance of understanding each step. By mastering these techniques, one can confidently tackle a wide range of quadratic equations and further enhance their algebraic skills.
Keywords
Factoring quadratic equations, solving quadratic equations, algebraic techniques, quadratic formula, standard form of quadratic equation, greatest common factor, perfect square trinomial, repeated roots, real solutions, binomial factors, rearranging equations, simplifying equations, step-by-step guide, mathematics, algebra.
