Solve $\sqrt{x+7}-3 \sqrt{2}=1$, leaving your answer in the form $m+n \sqrt{2}$, where $m$ and $n$ are integers.
Answer (1)
Isolate the square root: x + 7 = 1 + 3 2 .
Square both sides: x + 7 = ( 1 + 3 2 ) 2 .
Expand and simplify: x + 7 = 19 + 6 2 .
Solve for x : x = 12 + 6 2 .
12 + 6 2
Explanation
Understanding the Problem We are given the equation x + 7 − 3 2 = 1 . Our goal is to solve for x and express the answer in the form m + n 2 , where m and n are integers.
Isolating the Square Root First, let's isolate the square root term by adding 3 2 to both sides of the equation: x + 7 = 1 + 3 2
Squaring Both Sides Now, we square both sides of the equation to eliminate the square root: ( x + 7 ) 2 = ( 1 + 3 2 ) 2 This simplifies to: x + 7 = ( 1 + 3 2 ) 2
Expanding the Right Side Next, we expand the right side of the equation: x + 7 = 1 2 + 2 ( 1 ) ( 3 2 ) + ( 3 2 ) 2 x + 7 = 1 + 6 2 + 9 ( 2 ) x + 7 = 1 + 6 2 + 18 x + 7 = 19 + 6 2
Isolating x Now, we isolate x by subtracting 7 from both sides of the equation: x = 19 + 6 2 − 7
Simplifying for the Final Answer Finally, we simplify to find x : x = 12 + 6 2 So, x is in the form m + n 2 , where m = 12 and n = 6 .
Final Answer Therefore, the solution to the equation x + 7 − 3 2 = 1 is x = 12 + 6 2 .
Examples
Imagine you're building a garden and need to calculate the length of a diagonal support beam. If the sides of the right triangle formed by the beam are expressed with square roots, solving an equation like this helps you determine the exact length needed, ensuring a perfect fit and structural stability. This blend of algebra and geometry is crucial in construction and design, where precision is key for both aesthetics and safety.
