Question

√(5x² - 6x+8) - √(5x2 - 6x - 7) = 1.​

Answer 1

Answer:

$\sqrt{5 {x}^{2} - 6x + 8} \: - \sqrt{5 {x}^{2} - 6x - 7} = 1 \\ \sqrt{5 {x}^{2} - 6x + 8} = 1 + \sqrt{5 {x}^{2} - 6x - 7} \: \\squiring \: both \: sides \: \\ (\sqrt{5 {x}^{2} - 6x + 8}) ^{2} = (1 + \sqrt{5 {x}^{2} - 6x - 7})^{2} \: \\ 5 {x}^{2} - 6x + 8 = (1)^{2} + (5 {x}^{2} - 6x - 7)^{2} + 2(\sqrt{5 {x}^{2} - 6x - 7}) \: \\ 5 {x}^{2} - 6x + 8 =1 + 5 {x}^{2} - 6x - 7+ 2(\sqrt{5 {x}^{2} - 6x - 7}) \: \\ 8 = - 6 + 2\sqrt{5 {x}^{2} - 6x - 7} \: \\ 8 + 6 = 2\sqrt{5 {x}^{2} - 6x - 7} \: \\ 14 = 2\sqrt{5 {x}^{2} - 6x - 7} \: \\ \frac{14}{2} = \sqrt{5 {x}^{2} - 6x - 7} \: \\ 7 = \sqrt{5 {x}^{2} - 6x - 7} \: \\ squiring \: again \: both \: sides \\ 49 = 5 {x}^{2} - 6x - 7 \\ 0 = 5 {x}^{2} - 6x - 7 - 49 \\ 5 {x}^{2} - 6x - 56 = 0 \\ 5 {x}^{2} - 20x + 14x- 56 = 0 \\ 5x(x - 4) + 14(x - 4) = 0 \\ (x - 4)(5x + 14) = 0 \\ x - 4 = 0 \: or \: 5x + 14 = 0 \\ x = 4 \: or \: x = - \frac{14}{5}$