1) x - 3√x - 4 = 0x - 3√x = 4√x(x - 3) = 4Squaring both sides:x(x - 3)^2 = 16x(x^2 - 6x + 9) = 16x^3 - 6x^2 + 9x - 16 = 0
2) -x + 10√x - 21 = 010√x = x + 21√x = (x + 21)/10Squaring both sides:x = (x^2 + 42x + 441)/100100x = x^2 + 42x + 441x^2 - 58x + 441 = 0
3) 2x^2 - 5x + 1 = 0Using the quadratic formula:x = [5 ± √(5^2 - 4(2)(1))] / (2(2))x = [5 ± √(25 - 8)] / 4x = [5 ± √17] / 4
Therefore, the solutions to the given equations are:
1) x^3 - 6x^2 + 9x - 16 = 02) x^2 - 58x + 441 = 03) x = (5 ± √17) / 4
1) x - 3√x - 4 = 0
x - 3√x = 4
√x(x - 3) = 4
Squaring both sides:
x(x - 3)^2 = 16
x(x^2 - 6x + 9) = 16
x^3 - 6x^2 + 9x - 16 = 0
2) -x + 10√x - 21 = 0
10√x = x + 21
√x = (x + 21)/10
Squaring both sides:
x = (x^2 + 42x + 441)/100
100x = x^2 + 42x + 441
x^2 - 58x + 441 = 0
3) 2x^2 - 5x + 1 = 0
Using the quadratic formula:
x = [5 ± √(5^2 - 4(2)(1))] / (2(2))
x = [5 ± √(25 - 8)] / 4
x = [5 ± √17] / 4
Therefore, the solutions to the given equations are:
1) x^3 - 6x^2 + 9x - 16 = 0
2) x^2 - 58x + 441 = 0
3) x = (5 ± √17) / 4