CAT 2023 Slot 1 QA Question 4

The equation given is: √(5x + 9) + √(5x - 9) = 3(2 + √2)

And We find the value of √(10x + 9). The options are A) 3√31, B) 3√7, C) 2√7, D) 4√5. The correct answer is supposed to be B, but I need to work through it.

First, when dealing with equations involving square roots, a common strategy is to square both sides to eliminate the radicals. But since there are two square roots here, squaring once might not be enough. I might have to square twice, which can get a bit messy, but Taking it step by step.

Let the equation as: √(5x + 9) + √(5x - 9) = 3(2 + √2)

Let me call the left side as A for simplicity, so A = √(5x + 9) + √(5x - 9). Then, A = 3(2 + √2).

To eliminate the square roots, I'll square both sides: A² = [√(5x + 9) + √(5x - 9)]² = [3(2 + √2)]²

Expanding the left side: [√a + √b]² = a + b + 2√(ab) So, substituting a = 5x + 9 and b = 5x - 9: (5x + 9) + (5x - 9) + 2√[(5x + 9)(5x - 9)] = [3(2 + √2)]²

Simplify the left side: 5x + 9 + 5x - 9 = 10x So, left side becomes 10x + 2√[(5x + 9)(5x - 9)]

Now, the right side: [3(2 + √2)]² = 9(2 + √2)² First, compute (2 + √2)²: (2 + √2)² = 2² + 22√2 + (√2)² = 4 + 4√2 + 2 = 6 + 4√2 So, multiplying by 9: 9(6 + 4√2) = 54 + 36√2

Putting it all together: 10x + 2√[(5x + 9)(5x - 9)] = 54 + 36√2

Let me isolate the square root term: 2√[(5x + 9)(5x - 9)] = 54 + 36√2 - 10x

Divide both sides by 2: √[(5x + 9)(5x - 9)] = (54 + 36√2 - 10x)/2 Simplify the right side: 54/2 = 27, 36√2/2 = 18√2, 10x/2 = 5x So, √[(5x + 9)(5x - 9)] = 27 + 18√2 - 5x

Now, notice that (5x + 9)(5x - 9) is a difference of squares: (5x)^2 - (9)^2 = 25x² - 81

So, the equation becomes: √(25x² - 81) = 27 + 18√2 - 5x

Now, to eliminate this square root, I'll square both sides again: [√(25x² - 81)]² = (27 + 18√2 - 5x)² Which simplifies to: 25x² - 81 = (27 + 18√2 - 5x)²

Expanding the right side. Let C = 27 + 18√2, so the expression is (C - 5x)².

Expanding (C - 5x)²: C² - 2C5x + (5x)² = C² - 10C x + 25x²

So, substituting back C = 27 + 18√2: 25x² - 81 = (27 + 18√2)² - 10(27 + 18√2)x + 25x²

Wait, but on the left side we have 25x² - 81, and on the right side, we have 25x² plus other terms. Let me subtract 25x² from both sides to simplify: 25x² - 81 - 25x² = (27 + 18√2)² - 10(27 + 18√2)x + 25x² - 25x² Which simplifies to: -81 = (27 + 18√2)² - 10(27 + 18√2)x

Now, Computing (27 + 18√2)²: First, expand it: (27)^2 + 22718√2 + (18√2)^2 Compute each term: 27² = 729 22718√2 = 22718√2 = 972√2 (18√2)^2 = 3242 = 648 So, adding them up: 729 + 972√2 + 648 = (729 + 648) + 972√2 = 1377 + 972√2

So, the equation becomes: -81 = 1377 + 972√2 - 10(27 + 18√2)x

Let me rearrange terms to solve for x: -81 - 1377 - 972√2 = -10(27 + 18√2)x

Compute the left side: -81 - 1377 = -1458 So, -1458 - 972√2 = -10*(27 + 18