Solving For X When F(x) = X^2 - 2x And G(x) = 6x + 4 And (f + G)(x) = 0

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Understanding the Problem

In this mathematics problem, we are given two functions, f(x) and g(x), and asked to find the value of x for which the sum of these functions, (f + g)(x), equals zero. This involves understanding function notation, function addition, and solving quadratic equations. The functions provided are f(x) = x^2 - 2x and g(x) = 6x + 4. The goal is to find the x that satisfies the equation (f + g)(x) = 0. Essentially, we need to add the two functions together, set the resulting expression equal to zero, and then solve for x. This will likely involve combining like terms and potentially factoring or using the quadratic formula to find the roots of the equation.

The key concept here is function addition. When we add two functions, we are simply adding their respective expressions. So, (f + g)(x) means we take the expression for f(x) and add it to the expression for g(x). Once we have the combined expression, we set it equal to zero because we are looking for the x value where the combined function equals zero. The resulting equation will be a quadratic equation, given the presence of the x^2 term in f(x). Solving quadratic equations often involves factoring, using the quadratic formula, or completing the square. In this particular case, factoring might be the most straightforward approach, but if the quadratic expression doesn't factor easily, the quadratic formula is a reliable alternative. Therefore, the steps to solve this problem are: 1) add the functions f(x) and g(x), 2) set the resulting expression equal to zero, and 3) solve the quadratic equation for x. The solutions to the quadratic equation will be the values of x that satisfy the original condition (f + g)(x) = 0.

The final step is to identify which of the provided answer choices matches one of the solutions we found. This will give us the correct value of x for which the sum of the functions f(x) and g(x) is zero. It's important to double-check the solution by substituting it back into the original equation (f + g)(x) = 0 to ensure that it satisfies the equation. This helps to avoid any potential errors in the algebraic manipulation or equation-solving process. By systematically combining the functions, setting the result to zero, and solving for x, we can accurately determine the value of x that meets the given condition. The answer choices provided suggest that there is a specific solution within the options, and our calculation will confirm which of these options is indeed the correct value.

Solution Steps

  1. Function Addition: Our primary objective is to find the value of x for which (f + g)(x) = 0. The first step involves adding the two functions, f(x) and g(x). Given f(x) = x^2 - 2x and g(x) = 6x + 4, we add these together:

    (f + g)(x) = f(x) + g(x) = (x^2 - 2x) + (6x + 4).

    This step is crucial as it combines the two functions into a single expression that we can then set equal to zero. Function addition is a fundamental operation in function algebra, and understanding how to correctly add functions is essential for solving this type of problem. We are essentially combining the outputs of the two functions for the same input x. The resulting expression will be a new function that represents the sum of the two original functions. The next step involves simplifying this combined expression to make it easier to work with.

  2. Simplification: Next, we simplify the expression obtained in the previous step by combining like terms. We have:

    (f + g)(x) = x^2 - 2x + 6x + 4.

    Combining the x terms, -2x and 6x, gives us 4x. Therefore, the simplified expression is:

    (f + g)(x) = x^2 + 4x + 4.

    This simplification is essential for making the equation easier to solve. By combining like terms, we reduce the complexity of the expression, making it more manageable for the subsequent steps. In this case, we combined the terms involving x, which resulted in a simplified quadratic expression. Quadratic expressions are a common occurrence in algebra, and knowing how to manipulate and solve them is a crucial skill. This simplified expression is now ready to be set equal to zero, which is the next step in finding the value of x that satisfies the given condition.

  3. Setting to Zero: Now, we set the simplified expression equal to zero to find the values of x that satisfy the condition (f + g)(x) = 0:

    x^2 + 4x + 4 = 0.

    This step is the core of solving the problem. By setting the sum of the functions equal to zero, we are essentially finding the x-values where the combined function intersects the x-axis. These x-values are also known as the roots or zeros of the function. The resulting equation is a quadratic equation, which means it has a maximum of two solutions. Solving quadratic equations is a fundamental skill in algebra, and there are several methods to do so, including factoring, using the quadratic formula, and completing the square. In this particular case, the quadratic expression is a perfect square trinomial, which means it can be easily factored. Factoring is often the most straightforward method for solving quadratic equations when the expression is factorable, as it provides a direct path to finding the solutions. The next step involves factoring the quadratic expression to find the roots.

  4. Factoring: We can factor the quadratic expression x^2 + 4x + 4. Notice that it is a perfect square trinomial:

    x^2 + 4x + 4 = (x + 2)(x + 2) = (x + 2)^2.

    Factoring is a crucial step as it transforms the quadratic equation into a product of simpler expressions. When the product of these expressions is equal to zero, it means that at least one of the factors must be zero. In this case, the quadratic expression is a perfect square trinomial, which makes it particularly easy to factor. Recognizing and factoring perfect square trinomials is a valuable skill in algebra, as it simplifies the process of solving quadratic equations. The factored form of the equation, (x + 2)^2 = 0, indicates that we have a repeated root, meaning there is only one unique solution for x. This is because the factor (x + 2) appears twice. The next step involves setting the factor equal to zero and solving for x.

  5. Solving for x: Setting the factor equal to zero, we get:

    (x + 2) = 0.

    Solving for x, we subtract 2 from both sides:

    x = -2.

    This step is where we find the actual value of x that satisfies the original equation. By setting the factored expression equal to zero, we are essentially finding the roots of the quadratic equation. In this case, we have a repeated root, which means there is only one unique solution for x. The solution x = -2 is the value that makes the expression (x + 2) equal to zero, and therefore makes the entire quadratic expression equal to zero. This value of x is the solution to the equation (f + g)(x) = 0. It's important to check this solution by substituting it back into the original equation to ensure that it satisfies the equation. This is a good practice to avoid any potential errors in the algebraic manipulation or equation-solving process. The next step is to compare this solution with the given answer choices.

Answer

The value of x for which (f + g)(x) = 0 is x = -2. Comparing this with the given options, we see that it matches option B. -2.

Therefore, the correct answer is B. -2.

This final step confirms the solution by comparing it with the provided options. It's essential to make this comparison to ensure that the calculated answer matches one of the given choices. If the calculated answer doesn't match any of the options, it may indicate an error in the calculations, and it would be necessary to review the steps to identify and correct the error. In this case, the calculated solution x = -2 matches option B, which confirms that it is the correct answer. The entire process, from adding the functions to solving for x, has been carried out systematically, ensuring an accurate solution to the problem.