If $f(x) = X^2 - 2x$ And $g(x) = 6x + 4$, For Which Value Of $x$ Does $(f+g)(x) = 0$?A. \[$-4\$\]B. \[$-2\$\]C. 2D. 4

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Introduction

In mathematics, functions are used to describe relationships between variables. When we add two functions together, we get a new function that represents the sum of the two original functions. In this article, we will explore how to find the value of x for which the sum of two given functions equals zero.

Understanding the Given Functions

We are given two functions:

  • f(x)=x2−2xf(x) = x^2 - 2x
  • g(x)=6x+4g(x) = 6x + 4

To find the sum of these two functions, we need to add them together. This means we will combine like terms and simplify the resulting expression.

Finding the Sum of the Two Functions

To find the sum of the two functions, we add them together:

(f+g)(x)=f(x)+g(x)(f+g)(x) = f(x) + g(x)

(f+g)(x)=(x2−2x)+(6x+4)(f+g)(x) = (x^2 - 2x) + (6x + 4)

(f+g)(x)=x2−2x+6x+4(f+g)(x) = x^2 - 2x + 6x + 4

(f+g)(x)=x2+4x+4(f+g)(x) = x^2 + 4x + 4

Solving for the Value of x

Now that we have the sum of the two functions, we need to find the value of x for which this sum equals zero. To do this, we set the sum equal to zero and solve for x:

x2+4x+4=0x^2 + 4x + 4 = 0

Factoring the Quadratic Equation

To solve this quadratic equation, we can try to factor it. Unfortunately, this equation does not factor easily, so we will need to use other methods to solve it.

Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form ax2+bx+c=0ax^2 + bx + c = 0, the solutions are given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In our case, a=1a = 1, b=4b = 4, and c=4c = 4. Plugging these values into the quadratic formula, we get:

x=−4±42−4(1)(4)2(1)x = \frac{-4 \pm \sqrt{4^2 - 4(1)(4)}}{2(1)}

x=−4±16−162x = \frac{-4 \pm \sqrt{16 - 16}}{2}

x=−4±02x = \frac{-4 \pm \sqrt{0}}{2}

x=−42x = \frac{-4}{2}

x=−2x = -2

Conclusion

In this article, we explored how to find the value of x for which the sum of two given functions equals zero. We started by adding the two functions together and then set the resulting expression equal to zero. We used the quadratic formula to solve for x and found that the value of x is -2.

Final Answer

The final answer is −2\boxed{-2}.

Additional Information

It's worth noting that the quadratic equation x2+4x+4=0x^2 + 4x + 4 = 0 has a repeated root, which means that the graph of the function y=x2+4x+4y = x^2 + 4x + 4 touches the x-axis at the point where x = -2. This is because the discriminant of the quadratic equation is zero, which means that the quadratic equation has a repeated root.

Real-World Applications

The concept of finding the sum of two functions and solving for the value of x is used in many real-world applications, such as:

  • Physics: When modeling the motion of an object, we often need to find the sum of two or more functions to describe the object's position, velocity, or acceleration.
  • Engineering: In engineering, we often need to find the sum of two or more functions to describe the behavior of a system or a circuit.
  • Economics: In economics, we often need to find the sum of two or more functions to describe the behavior of a market or an economy.

Common Mistakes

When solving for the value of x, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Not simplifying the expression: Make sure to simplify the expression before solving for x.
  • Not using the correct formula: Make sure to use the correct formula for solving quadratic equations.
  • Not checking the solutions: Make sure to check the solutions to make sure they are valid.

Conclusion

In conclusion, finding the sum of two functions and solving for the value of x is an important concept in mathematics. By following the steps outlined in this article, you can solve for the value of x and apply this concept to real-world problems.

Q: What is the sum of two functions?

A: The sum of two functions is a new function that represents the combination of the two original functions. To find the sum of two functions, we add them together by combining like terms and simplifying the resulting expression.

Q: How do I find the sum of two functions?

A: To find the sum of two functions, we add them together by combining like terms and simplifying the resulting expression. For example, if we have two functions:

  • f(x)=x2−2xf(x) = x^2 - 2x
  • g(x)=6x+4g(x) = 6x + 4

We can find the sum of these two functions by adding them together:

(f+g)(x)=f(x)+g(x)(f+g)(x) = f(x) + g(x)

(f+g)(x)=(x2−2x)+(6x+4)(f+g)(x) = (x^2 - 2x) + (6x + 4)

(f+g)(x)=x2−2x+6x+4(f+g)(x) = x^2 - 2x + 6x + 4

(f+g)(x)=x2+4x+4(f+g)(x) = x^2 + 4x + 4

Q: How do I solve for the value of x in the sum of two functions?

A: To solve for the value of x in the sum of two functions, we need to set the sum equal to zero and solve for x. We can use the quadratic formula to solve for x:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form ax2+bx+c=0ax^2 + bx + c = 0, the solutions are given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I use the quadratic formula to solve for x?

A: To use the quadratic formula to solve for x, we need to plug in the values of a, b, and c into the formula. For example, if we have the equation:

x2+4x+4=0x^2 + 4x + 4 = 0

We can plug in the values a = 1, b = 4, and c = 4 into the quadratic formula:

x=−4±42−4(1)(4)2(1)x = \frac{-4 \pm \sqrt{4^2 - 4(1)(4)}}{2(1)}

x=−4±16−162x = \frac{-4 \pm \sqrt{16 - 16}}{2}

x=−4±02x = \frac{-4 \pm \sqrt{0}}{2}

x=−42x = \frac{-4}{2}

x=−2x = -2

Q: What is the final answer?

A: The final answer is −2\boxed{-2}.

Q: What are some common mistakes to avoid when solving for the value of x?

A: Some common mistakes to avoid when solving for the value of x include:

  • Not simplifying the expression: Make sure to simplify the expression before solving for x.
  • Not using the correct formula: Make sure to use the correct formula for solving quadratic equations.
  • Not checking the solutions: Make sure to check the solutions to make sure they are valid.

Q: What are some real-world applications of finding the sum of two functions and solving for the value of x?

A: Some real-world applications of finding the sum of two functions and solving for the value of x include:

  • Physics: When modeling the motion of an object, we often need to find the sum of two or more functions to describe the object's position, velocity, or acceleration.
  • Engineering: In engineering, we often need to find the sum of two or more functions to describe the behavior of a system or a circuit.
  • Economics: In economics, we often need to find the sum of two or more functions to describe the behavior of a market or an economy.

Q: What are some additional tips for solving for the value of x?

A: Some additional tips for solving for the value of x include:

  • Make sure to check your work: Make sure to check your work to make sure you have the correct solution.
  • Use a calculator: If you are having trouble solving for x, try using a calculator to check your work.
  • Ask for help: If you are having trouble solving for x, don't be afraid to ask for help.