Math Problem Statement
Find all real solutions of the equation. (Enter your answers as a comma-separated list. If there is no real solution, enter NO REAL SOLUTION.) 2x + root ( x + 6)= 9 x =
Solution
To solve the equation , let's follow these steps:
Step 1: Isolate the square root term
First, isolate the square root term by subtracting from both sides of the equation:
Step 2: Square both sides
Square both sides of the equation to eliminate the square root:
Step 3: Expand the square on the right-hand side
Now, expand :
Step 4: Rearrange the equation
Rearrange the equation into a standard quadratic form:
Step 5: Solve the quadratic equation
Use the quadratic formula , where , , and :
Step 6: Find the two possible values of
Step 7: Check the solutions in the original equation
Now, substitute these values back into the original equation to check if they are valid:
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For : This does not satisfy the original equation, so is not a solution.
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For : This satisfies the original equation, so is a valid solution.
Final Answer:
The only real solution is:
Would you like a further breakdown or have any questions?
Related Questions:
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Tip: Always check your solutions in the original equation, especially when dealing with square roots or other powers.
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Math Problem Analysis
Mathematical Concepts
Equations
Quadratic Equations
Square Roots
Formulas
Quadratic formula
Theorems
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Suitable Grade Level
Grades 9-12