Solve for all values of x: sqrt (4x)-2=sqrt (9x+7) Answer Attemptiout of 2 (A) Additional Solution (C) No Solution x=

To solve the equation $\sqrt{4x} - 2 = \sqrt{9x + 7}$, we need to isolate the square roots and then square both sides to eliminate the square roots.<br /><br />Step 1: Isolate one of the square roots.<br />$\sqrt{4x} - 2 = \sqrt{9x + 7}$<br />$\sqrt{4x} = \sqrt{9x + 7} + 2$<br /><br />Step 2: Square both sides of the equation.<br />$(\sqrt{4x})^2 = (\sqrt{9x + 7} + 2)^2$<br />$4x = (9x + 7) + 4\sqrt{9x + 7} + 4$<br />$4x = 9x + 11 + 4\sqrt{9x + 7}$<br /><br />Step 3: Isolate the remaining square root.<br />$4x - 9x = 11 + 4\sqrt{9x + 7}$<br />$-5x = 11 + 4\sqrt{9x + 7}$<br />$-5x - 11 = 4\sqrt{9x + 7}$<br />$\frac{-5x - 11}{4} = \sqrt{9x + 7}$<br /><br />Step 4: Square both sides of the equation again.<br />$\left(\frac{-5x - 11}{4}\right)^2 = (\sqrt{9x + 7})^2$<br />$\frac{25x^2 + 110x + 121}{16} = 9x + 7$<br />$25x^2 + 110x + 121 = 144x + 112$<br />$25x^2 - 34x + 9 = 0$<br /><br />Step 5: Solve the quadratic equation.<br />Using the quadratic formula, we can solve for x:<br />$x = \frac{-(-34) \pm \sqrt{(-34)^2 - 4(25)(9)}}{2(25)}$<br />$x = \frac{34 \pm \sqrt{1156 - 900}}{50}$<br />$x = \frac{34 \pm \sqrt{256}}{50}$<br />$x = \frac{34 \pm 16}{50}$<br /><br />Therefore, the solutions for x are:<br />$x = \frac{50}{50} = 1$<br />$x = \frac{18}{50} = \frac{9}{25}$<br /><br />So, the correct answer is (A) Additional Solution, and the values of x are $x = 1$ and $x = \frac{9}{25}$.