Solve By Factoring 2x - 11√x + 12 = 0 A Step-by-Step Guide
Factoring is a powerful technique in algebra used to simplify and solve equations. When faced with equations involving fractional exponents or radicals, factoring can still be an effective method, albeit one that requires careful manipulation and substitution. In this article, we will delve into the process of solving the equation 2x - 11√x + 12 = 0 by factoring. This equation is a classic example of a quadratic-like equation, which can be transformed into a standard quadratic equation through a clever substitution. We will explore each step in detail, ensuring a clear understanding of the underlying principles.
Understanding the Equation
Before we jump into the solution, let's first dissect the equation 2x - 11√x + 12 = 0. This equation might not immediately strike you as a quadratic equation, but it shares a similar structure. Notice the presence of three terms: a term with 'x', a term with the square root of 'x' (√x), and a constant term. This structure hints at the possibility of transforming the equation into a quadratic form using a suitable substitution.
The key observation here is that x can be expressed as (√x)². This relationship allows us to rewrite the equation in terms of √x, paving the way for a quadratic substitution. By recognizing this underlying structure, we can leverage the well-established techniques for solving quadratic equations, such as factoring, to find the solutions for our original equation. Understanding the relationship between the terms is crucial for choosing the correct substitution and simplifying the equation effectively. The goal is to transform the equation into a form that is easier to manipulate and solve, and recognizing the quadratic-like structure is the first step in achieving this.
Making a Substitution
The heart of solving this equation lies in recognizing its quadratic form. To make this clearer, we introduce a substitution. Let's set y = √x. This substitution is crucial because it transforms the original equation into a standard quadratic equation, which we can then solve using familiar methods. When we square both sides of the substitution, we get y² = x. This relationship is essential for rewriting the original equation in terms of 'y'.
Now, substituting 'y' into the original equation 2x - 11√x + 12 = 0, we replace 'x' with 'y²' and '√x' with 'y'. This transformation yields the new equation: 2y² - 11y + 12 = 0. This equation is now a standard quadratic equation in the variable 'y'. The beauty of this substitution is that it simplifies the equation's structure, making it easier to factor and solve. By replacing the more complex terms with a single variable, we can apply the well-known techniques for solving quadratic equations, such as factoring, the quadratic formula, or completing the square. The substitution is a powerful tool for simplifying equations and revealing their underlying structure.
Factoring the Quadratic Equation
Now that we have the quadratic equation 2y² - 11y + 12 = 0, we can proceed to factor it. Factoring involves expressing the quadratic expression as a product of two binomials. To factor this equation, we need to find two numbers that multiply to the product of the leading coefficient (2) and the constant term (12), which is 24, and add up to the middle coefficient (-11). These two numbers are -8 and -3, since (-8) * (-3) = 24 and (-8) + (-3) = -11.
Using these numbers, we can rewrite the middle term of the quadratic equation: 2y² - 8y - 3y + 12 = 0. This step is crucial for the factoring process, as it allows us to group terms and factor by grouping. Next, we group the terms in pairs: (2y² - 8y) + (-3y + 12) = 0. From the first group, we can factor out 2y, and from the second group, we can factor out -3: 2y(y - 4) - 3(y - 4) = 0. Notice that both terms now have a common factor of (y - 4). We can factor this out to obtain the factored form of the quadratic equation: (2y - 3)(y - 4) = 0. This factored form is equivalent to the original quadratic equation and represents the product of two binomials that equal zero. This is a significant step, as it allows us to apply the zero-product property to find the solutions for 'y'.
Applying the Zero-Product Property
The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. This property is a cornerstone of solving equations by factoring. In our case, we have the factored equation (2y - 3)(y - 4) = 0. According to the zero-product property, this equation holds true if either 2y - 3 = 0 or y - 4 = 0.
We now have two simple linear equations to solve. Let's solve the first equation, 2y - 3 = 0. To isolate 'y', we add 3 to both sides of the equation: 2y = 3. Then, we divide both sides by 2 to get y = 3/2. This is one possible solution for 'y'.
Next, we solve the second equation, y - 4 = 0. To isolate 'y', we add 4 to both sides of the equation: y = 4. This is another possible solution for 'y'. So, we have found two solutions for 'y': y = 3/2 and y = 4. These solutions are crucial, but remember that we are solving for 'x', not 'y'. We need to substitute back to find the corresponding values of 'x'. The zero-product property is a powerful tool that allows us to break down a factored equation into simpler equations, making it easier to find the solutions.
Substituting Back to Find x
We have found the solutions for 'y', but our original equation is in terms of 'x'. Recall our substitution: y = √x. To find the values of 'x', we need to substitute back our solutions for 'y' into this equation. Let's start with the first solution, y = 3/2. Substituting this into y = √x, we get 3/2 = √x. To solve for 'x', we square both sides of the equation: (3/2)² = (√x)², which simplifies to 9/4 = x. So, one solution for 'x' is x = 9/4.
Now, let's consider the second solution, y = 4. Substituting this into y = √x, we get 4 = √x. Again, we square both sides of the equation to solve for 'x': 4² = (√x)², which simplifies to 16 = x. Thus, another solution for 'x' is x = 16. We have now found two potential solutions for 'x': x = 9/4 and x = 16. However, it's crucial to verify these solutions by plugging them back into the original equation to ensure they are valid.
Verifying the Solutions
In algebra, it's essential to verify the solutions we obtain, especially when dealing with radical equations. This is because the process of squaring both sides of an equation can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original one. To verify our solutions, we will plug them back into the original equation 2x - 11√x + 12 = 0.
Let's start with the first solution, x = 9/4. Substituting this value into the equation, we get: 2(9/4) - 11√(9/4) + 12 = 0. Simplifying this, we have: 9/2 - 11(3/2) + 12 = 0. Further simplification gives us: 9/2 - 33/2 + 12 = 0. Combining the fractions, we get: -24/2 + 12 = 0, which simplifies to -12 + 12 = 0. This confirms that x = 9/4 is a valid solution.
Now, let's verify the second solution, x = 16. Substituting this value into the equation, we get: 2(16) - 11√16 + 12 = 0. Simplifying this, we have: 32 - 11(4) + 12 = 0. Further simplification gives us: 32 - 44 + 12 = 0, which simplifies to 0 = 0. This confirms that x = 16 is also a valid solution. Since both solutions satisfy the original equation, we can confidently state that our solutions are correct.
Final Answer
After meticulously solving the equation 2x - 11√x + 12 = 0 by factoring and verifying our solutions, we arrive at the final answer. We began by recognizing the quadratic-like structure of the equation and making a suitable substitution, y = √x. This substitution transformed the equation into a standard quadratic equation, which we then factored. By applying the zero-product property, we found two solutions for 'y'. Substituting back, we obtained two potential solutions for 'x': x = 9/4 and x = 16.
Finally, we verified these solutions by plugging them back into the original equation, confirming that both solutions are valid. Therefore, the solutions to the equation 2x - 11√x + 12 = 0 are x = 9/4 and x = 16. This process illustrates the power of factoring and substitution in solving complex equations. By breaking down the problem into manageable steps and carefully verifying our solutions, we can confidently arrive at the correct answer.
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Solve by Factoring 2x - 11√x + 12 = 0 A Step-by-Step Guide
