Solve For \[$ X \$\] In The Equation \[$ H(x) = 4 \$\].Given: \[$ H(4) = X^2 = 5 \$\].

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Solving for x in the Equation h(x) = 4

In this article, we will delve into solving for x in the equation h(x) = 4, given that h(4) = x^2 = 5. This problem involves algebraic manipulation and substitution to find the value of x that satisfies the equation.

Understanding the Equation

The equation h(x) = 4 represents a function h(x) that takes an input x and produces an output of 4. In other words, for any value of x, the function h(x) will always return 4. The given equation h(4) = x^2 = 5 provides additional information about the function h(x) when the input is 4.

Breaking Down the Given Equation

The equation h(4) = x^2 = 5 can be broken down into two parts:

  • h(4) = 5: This equation tells us that when the input to the function h(x) is 4, the output is 5.
  • x^2 = 5: This equation tells us that the square of x is equal to 5.

Solving for x

To solve for x, we need to find the value of x that satisfies the equation x^2 = 5. We can do this by taking the square root of both sides of the equation.

Taking the Square Root

When we take the square root of both sides of the equation x^2 = 5, we get:

x = ±√5

Understanding the ± Symbol

The ± symbol indicates that there are two possible values of x that satisfy the equation. The positive value of x is √5, and the negative value of x is -√5.

Substituting x into the Original Equation

Now that we have found the values of x, we can substitute them into the original equation h(x) = 4 to verify that they satisfy the equation.

Verifying the Solution

Substituting x = √5 into the original equation h(x) = 4, we get:

h(√5) = 4

This equation is true, since the function h(x) always returns 4.

Similarly, substituting x = -√5 into the original equation h(x) = 4, we get:

h(-√5) = 4

This equation is also true, since the function h(x) always returns 4.

In conclusion, we have solved for x in the equation h(x) = 4, given that h(4) = x^2 = 5. We found that x = ±√5, and verified that these values satisfy the original equation. This problem demonstrates the importance of algebraic manipulation and substitution in solving equations.

Additional Tips and Variations

  • If the equation h(x) = 4 had been given as h(x) = x^2 = 4, the solution would have been x = ±2.
  • If the equation h(x) = 4 had been given as h(x) = x^2 + 1 = 4, the solution would have been x = ±√3.
  • If the equation h(x) = 4 had been given as h(x) = x^2 - 5 = 4, the solution would have been x = ±3.

Solving for x in the equation h(x) = 4, given that h(4) = x^2 = 5, requires careful algebraic manipulation and substitution. By following the steps outlined in this article, we can find the value of x that satisfies the equation. This problem is a great example of how algebra can be used to solve real-world problems.
Solving for x in the Equation h(x) = 4: Q&A

In our previous article, we solved for x in the equation h(x) = 4, given that h(4) = x^2 = 5. In this article, we will answer some frequently asked questions about the problem and provide additional insights.

Q: What is the function h(x)?

A: The function h(x) is a mathematical function that takes an input x and produces an output of 4. In other words, for any value of x, the function h(x) will always return 4.

Q: What does the equation h(4) = x^2 = 5 mean?

A: The equation h(4) = x^2 = 5 tells us that when the input to the function h(x) is 4, the output is 5. It also tells us that the square of x is equal to 5.

Q: How do we solve for x in the equation h(x) = 4?

A: To solve for x, we need to find the value of x that satisfies the equation x^2 = 5. We can do this by taking the square root of both sides of the equation.

Q: What are the possible values of x?

A: The possible values of x are x = ±√5. The positive value of x is √5, and the negative value of x is -√5.

Q: How do we verify that the values of x satisfy the original equation?

A: We can verify that the values of x satisfy the original equation by substituting them into the equation h(x) = 4. If the equation is true, then the values of x are valid solutions.

Q: What if the equation h(x) = 4 had been given as h(x) = x^2 = 4?

A: If the equation h(x) = 4 had been given as h(x) = x^2 = 4, the solution would have been x = ±2.

Q: What if the equation h(x) = 4 had been given as h(x) = x^2 + 1 = 4?

A: If the equation h(x) = 4 had been given as h(x) = x^2 + 1 = 4, the solution would have been x = ±√3.

Q: What if the equation h(x) = 4 had been given as h(x) = x^2 - 5 = 4?

A: If the equation h(x) = 4 had been given as h(x) = x^2 - 5 = 4, the solution would have been x = ±3.

Q: Can we use algebraic manipulation to solve other types of equations?

A: Yes, algebraic manipulation can be used to solve other types of equations. The key is to identify the type of equation and use the appropriate techniques to solve it.

In conclusion, solving for x in the equation h(x) = 4, given that h(4) = x^2 = 5, requires careful algebraic manipulation and substitution. By following the steps outlined in this article, we can find the value of x that satisfies the equation. This problem is a great example of how algebra can be used to solve real-world problems.

Additional Tips and Resources

  • For more information on algebraic manipulation, see our article on "Algebraic Manipulation: A Guide to Solving Equations".
  • For more practice problems, see our article on "Algebra Practice Problems: A Collection of Equations to Solve".
  • For more resources on algebra, see our article on "Algebra Resources: A Collection of Online Resources".