250/MR00. problem / Math - Solved Math Problems

MR-250. problem

Source: MathReference: Generic Math Textbook - 250.)

Algebra → Exponential equations and inequalities → Exponential equations

\sqrt{x^{2} - \sqrt{4 x^{2} - 11}} = x - 1

\sqrt{x^{2} - \sqrt{4 x^{2} - 11}} = x - 1, \Rightarrow 4 x^{2} - 11 > 0

4 x^{2} > 11, x^{2} > \frac{11}{4}, x > \frac{\sqrt{11}}{2}, x < - \frac{\sqrt{11}}{2}

{(\sqrt{x^{2} - \sqrt{4 x^{2} - 11}})}^{2} = {(x - 1)}^{2}

x^{2} - \sqrt{4 x^{2} - 11} = x^{2} - 2 x + 1

x^{2} - x^{2} + 2 x - 1 = \sqrt{4 x^{2} - 11}

2 x - 1 = \sqrt{4 x^{2} - 11}

{(2 x - 1)}^{2} = {(\sqrt{4 x^{2} - 11})}^{2}

4 x^{2} - 4 x + 1 = 4 x^{2} - 11

4 x^{2} - 4 x^{2} = 4 x - 1 - 11

0 = 4 x - 12

12 = 4 x

x = 3 > \frac{\sqrt{11}}{2} = 1, 658

x = 3

x = 3