Math Problem Statement
2x
Solution
We are given the equation:
Step 1: Substitute a new variable
Let's simplify this by substituting , so and .
Substitute these into the equation:
Step 2: Simplify the equation
Now, simplify the terms:
Expand the first term:
Step 3: Eliminate the fraction
Multiply the entire equation by to eliminate the denominator:
Step 4: Combine like terms
Simplify the quadratic expression:
Step 5: Solve for
Rearrange the equation:
Step 6: Solve for
Take the square root of both sides:
Step 7: Solve for
Recall that . Substituting :
Square both sides:
Solve for :
Thus, the solution is:
Let me know if you want more details or further explanation.
Here are 5 related questions:
- How does substitution help simplify complex equations?
- What is the importance of eliminating fractions in an equation?
- Why do we square both sides when solving equations involving square roots?
- What does it mean for a solution to have both positive and negative square roots?
- How can we verify the solution by substituting it back into the original equation?
Tip: When solving equations involving square roots, always check if both the positive and negative roots provide valid solutions in the context of the problem.
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Math Problem Analysis
Mathematical Concepts
Substitution
Algebra
Square Roots
Quadratic Equations
Formulas
y = \sqrt{4 - x}, y^2 = 4 - x
Quadratic equation: ax^2 + bx + c = 0
Solving square roots: \(y = \pm \frac{2\sqrt{10}}{5}\)
Theorems
Substitution method for simplifying complex equations
Square root theorem for solving quadratic expressions
Suitable Grade Level
Grade 10-12