Solve: [tex]$\sqrt{x+3}=\sqrt{2 x-3}$[/tex]
Answer (2)
To solve the equation x + 3 = 2 x − 3 , square both sides to eliminate the square roots, leading to x + 3 = 2 x − 3 . By isolating x and solving, we find that x = 6 , which is confirmed by substituting back into the original equation. Therefore, the solution is x = 6 .
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Square both sides of the equation to get rid of the square roots: x + 3 = 2 x − 3 .
Isolate x by subtracting x from both sides: 3 = x − 3 .
Solve for x by adding 3 to both sides: x = 6 .
Verify the solution in the original equation, which confirms that x = 6 is the correct answer: 6 .
Explanation
Understanding the Problem We are given the equation x + 3 = 2 x − 3 . Our goal is to find the value of x that satisfies this equation.
Eliminating Square Roots To eliminate the square roots, we square both sides of the equation: ( x + 3 ) 2 = ( 2 x − 3 ) 2 This simplifies to: x + 3 = 2 x − 3
Isolating x Now, we want to isolate x . Subtract x from both sides of the equation: x + 3 − x = 2 x − 3 − x This simplifies to: 3 = x − 3
Solving for x Next, add 3 to both sides of the equation to solve for x : 3 + 3 = x − 3 + 3 This gives us: 6 = x So, x = 6 .
Checking the Solution We need to check if our solution is valid by substituting x = 6 back into the original equation: 6 + 3 = 2 ( 6 ) − 3 9 = 12 − 3 9 = 9 Since both sides are equal, our solution is valid.
Final Answer Therefore, the solution to the equation x + 3 = 2 x − 3 is x = 6 .
Examples
Imagine you're designing a garden and need to ensure two paths have the same length, represented by square root expressions. Solving such an equation helps determine the exact point where the paths meet, ensuring symmetry and balance in your garden design. This principle applies in various fields, from architecture to physics, where maintaining equality between expressions is crucial for accurate designs and predictions.
