Solve For All Possible Values Of X X X . 8 X − 4 = X + 1 \sqrt{8x - 4} = X + 1 8 X − 4 ​ = X + 1

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Understanding the Problem

The given equation is a radical equation, which involves a square root. The goal is to solve for all possible values of xx that satisfy the equation 8x4=x+1\sqrt{8x - 4} = x + 1. This equation involves a quadratic expression inside the square root, making it a bit more complex than a simple linear or quadratic equation.

Step 1: Square Both Sides of the Equation

To eliminate the square root, we can square both sides of the equation. This will help us get rid of the radical sign and make it easier to solve for xx. Squaring both sides gives us:

(8x4)2=(x+1)2\left(\sqrt{8x - 4}\right)^2 = (x + 1)^2

Step 2: Expand and Simplify the Equation

Expanding the squared expressions on both sides, we get:

8x4=x2+2x+18x - 4 = x^2 + 2x + 1

Now, let's simplify the equation by moving all terms to one side:

x26x+5=0x^2 - 6x + 5 = 0

Step 3: Factor the Quadratic Equation

The quadratic equation x26x+5=0x^2 - 6x + 5 = 0 can be factored as:

(x5)(x1)=0(x - 5)(x - 1) = 0

Step 4: Solve for xx

To find the values of xx, we can set each factor equal to zero and solve for xx:

x5=0x=5x - 5 = 0 \Rightarrow x = 5

x1=0x=1x - 1 = 0 \Rightarrow x = 1

Step 5: Check the Solutions

Before accepting the solutions, we need to check if they satisfy the original equation. Let's plug in x=5x = 5 and x=1x = 1 into the original equation:

8(5)4=5+1404=636=66=6\sqrt{8(5) - 4} = 5 + 1 \Rightarrow \sqrt{40 - 4} = 6 \Rightarrow \sqrt{36} = 6 \Rightarrow 6 = 6

8(1)4=1+144=20=20=2\sqrt{8(1) - 4} = 1 + 1 \Rightarrow \sqrt{4 - 4} = 2 \Rightarrow \sqrt{0} = 2 \Rightarrow 0 = 2

The second solution, x=1x = 1, does not satisfy the original equation, so we can discard it.

Conclusion

The only solution to the equation 8x4=x+1\sqrt{8x - 4} = x + 1 is x=5x = 5. This is the only value of xx that satisfies the original equation.

Final Answer

The final answer is 5\boxed{5}.

Related Topics

  • Solving quadratic equations
  • Radical equations
  • Algebraic manipulations

Additional Resources

  • Khan Academy: Solving Quadratic Equations
  • Mathway: Radical Equations
  • Wolfram Alpha: Algebraic Manipulations

Understanding Radical Equations

Radical equations are equations that involve a square root or other radical expression. They can be more challenging to solve than linear or quadratic equations, but with the right techniques and strategies, you can master them. In this article, we'll answer some common questions about solving radical equations.

Q: What is a radical equation?

A: A radical equation is an equation that involves a square root or other radical expression. It can be written in the form a=b\sqrt{a} = b, where aa and bb are expressions involving variables.

Q: How do I solve a radical equation?

A: To solve a radical equation, you can follow these steps:

  1. Square both sides of the equation to eliminate the radical sign.
  2. Simplify the resulting equation by combining like terms.
  3. Solve for the variable using algebraic manipulations.
  4. Check the solutions to make sure they satisfy the original equation.

Q: What are some common mistakes to avoid when solving radical equations?

A: Some common mistakes to avoid when solving radical equations include:

  • Squaring both sides of the equation without checking if the expression inside the square root is non-negative.
  • Failing to simplify the resulting equation after squaring both sides.
  • Not checking the solutions to make sure they satisfy the original equation.

Q: How do I know if the expression inside the square root is non-negative?

A: To determine if the expression inside the square root is non-negative, you can use the following rules:

  • If the expression inside the square root is a polynomial, you can check if it has any negative coefficients.
  • If the expression inside the square root is a rational expression, you can check if the numerator is non-negative and the denominator is positive.
  • If the expression inside the square root is a trigonometric expression, you can check if the argument of the trigonometric function is within the correct range.

Q: What are some examples of radical equations?

A: Some examples of radical equations include:

  • x+1=2\sqrt{x + 1} = 2
  • 2x3=x1\sqrt{2x - 3} = x - 1
  • x2+1=x+2\sqrt{x^2 + 1} = x + 2

Q: How do I solve a radical equation with a quadratic expression inside the square root?

A: To solve a radical equation with a quadratic expression inside the square root, you can follow these steps:

  1. Square both sides of the equation to eliminate the radical sign.
  2. Simplify the resulting equation by combining like terms.
  3. Factor the quadratic expression inside the square root.
  4. Solve for the variable using algebraic manipulations.
  5. Check the solutions to make sure they satisfy the original equation.

Q: What are some real-world applications of radical equations?

A: Radical equations have many real-world applications, including:

  • Physics: Radical equations are used to model the motion of objects under the influence of gravity or other forces.
  • Engineering: Radical equations are used to design and optimize systems, such as bridges or buildings.
  • Computer Science: Radical equations are used in algorithms for solving problems, such as finding the shortest path between two points.

Conclusion

Radical equations can be challenging to solve, but with the right techniques and strategies, you can master them. By following the steps outlined in this article, you can solve radical equations and apply them to real-world problems.

Final Answer

The final answer is 5\boxed{5}.

Related Topics

  • Solving quadratic equations
  • Algebraic manipulations
  • Real-world applications of mathematics

Additional Resources

  • Khan Academy: Solving Radical Equations
  • Mathway: Radical Equations
  • Wolfram Alpha: Algebraic Manipulations