There is another strong method, that can save you some time in the cases with simple answers: Monotony. √(x) - increases monotonically (*where it exists*) √(x-9) - increases monotonically √(x) + √(x-9) - increases monotonically 9 - constant So the left side is increasing monotonically, while the right side is a constant => no more than one solution. I guess x = 25. Oh, it's right, and that is the only answer, because of monotony.
Here's how I did it. The equation is equivalent to the system: a+b=9 a^2-b^2=9, where a=sqrt(x) But: a^2-b^2=(a-b)(a+b)=9(a-b)=9 So, a-b=1. Summing it with a+b=9, we get 2a=10. So a=5, which means sqrt(x)=5, and x=25
Par analyse-synthèse : soit (E) l'équation donnée, alors si x (un réel) est solution de (E), on a : sqrt(x) + sqrt(x-9) = 9 x + 2sqrt(x)sqrt(x-9) + x-9 = 81 (en mettant au carré) 2x + 2sqrt(x(x-9)) = 90 x + sqrt(x^2 - 9x) = 45 (en divisant par deux) x^2 - 9x = 2025 - 90x + x^2 (en développant) 81x = 2025 x = 2025/81 = 25 (mental calculation: 81*25 = 80*25 [=1600+400=2000] + 25 = 2025) Synthèse : On a sqrt(x) et sqrt(x-9) dans l'équation initiale, donc x>0 et x-9>0, donc x>9. Or le seul candidat, 25, est >9. Donc l'unique solution est 25. This is the mathematically rigorous way to do it.
If you can use casio, this could be easier by using solution to create a multiplier. Like 2x+x^2=x^3 solution is 2 so we can do this: 2x-4+x^2-4=x^3-8 2(x-2)+(x-2)(x+2)=(x-2)(x^2+2x+4) x=2 and proving 2+x+2=x^2+2x+4 have solution or not.
Was scrolling this on my phone in bed, and solved it from the thumbnail in my head. Although I did use the calculator app for my last step to solve 324x=8100
Is pretty simple actually there's Just one perfect square that IS exactly 9 bigger than another ,is 25 ,that IS 5² ,that IS 9 bigger than 4² and 5+4 is equal 9
I saw the preview and tried to solve it completely on myself. But my solution is incorrect and I dunno why... Can someone tell what did I do wrong? 😢 √(x) + √(x-9) = 9 √(x) - 9 + √(x-9) = 0 (√(√x))² - 3² + √(x-9) = 0 (√(x) + 3)(√(x) - 3) + √(x-9) = 0 x + 3√(x) - 3√(x) - 9 + √(x-9) = 0 x - 9 = √(x-9) (x-9)² = (√(x-9))² x² - 18x + 81 = x - 9 x² - 19x + 90 = 0 a=1, b=-19, c=90 D = 361 - 360 = 1 x1 = (19+1)/2 = 10 x2 = 18/2 = 9 I checked this solution few times, it must be correct...
Your mistake is between steps 3 and 4. When you expanded your a^2-b^2, one of your square roots disappeared. A is the quartic root of x, not the square root
