Solve for all values of x: sqrt (x)+4=sqrt (5x+4) Answer Attemptiont of Additional Solution (C) No Solution x=

To solve the equation $\sqrt{x} + 4 = \sqrt{5x + 4}$, we can follow these steps:<br /><br />1. Isolate one of the square roots on one side of the equation:<br /> $\sqrt{x} = \sqrt{5x + 4} - 4$<br /><br />2. Square both sides of the equation to eliminate the square root:<br /> $(\sqrt{x})^2 = (\sqrt{5x + 4} - 4)^2$<br /> $x = (5x + 4) - 8\sqrt{5x + 4} + 16$<br /> $x = 5x + 20 - 8\sqrt{5x + 4}$<br /><br />3. Isolate the square root term:<br /> $x - 5x = 20 - 8\sqrt{5x + 4}$<br /> $-4x = 20 - 8\sqrt{5x + 4}$<br /> $8\sqrt{5x + 4} = 20 - 4x$<br /> $\sqrt{5x + 4} = \frac{20 - 4x}{8}$<br /> $\sqrt{5x + 4} = \frac{5 - x}{2}$<br /><br />4. Square both sides of the equation again to eliminate the square root:<br /> $(\sqrt{5x + 4})^2 = (\frac{5 - x}{2})^2$<br /> $5x + 4 = \frac{(5 - x)^2}{4}$<br /> $20x + 16 = (5 - x)^2$<br /> $20x + 16 = 25 - 10x + x^2$<br /> $x^2 - 30x + 9 = 0$<br /><br />5. Solve the quadratic equation:<br /> $x^2 - 30x + 9 = 0$<br /> $x = \frac{30 \pm \sqrt{900 - 36}}{2}$<br /> $x = \frac{30 \pm \sqrt{864}}{2}$<br /> $x = \frac{30 \pm 29.39}{2}$<br /><br />Therefore, the solutions to the equation $\sqrt{x} + 4 = \sqrt{5x + 4}$ are:<br />$x = \frac{30 + 29.39}{2} \approx 29.695$<br />$x = \frac{30 - 29.39}{2} \approx 0.305$<br /><br />So, the final answer is:<br />$x = 29.695$ or $x = 0.305$