Solve for all values of x: sqrt (x)-5=sqrt (x+6) Answer Attempt 1 out of 2 (4) Additional Solution No Solution z=

To solve the equation $\sqrt{x} - 5 = \sqrt{x+6}$, we can start by isolating one of the square roots.<br /><br />First, let's isolate $\sqrt{x}$ by adding 5 to both sides of the equation:<br /><br />$\sqrt{x} = \sqrt{x+6} + 5$<br /><br />Next, we can square both sides of the equation to eliminate the square roots:<br /><br />$(\sqrt{x})^2 = (\sqrt{x+6} + 5)^2$<br /><br />Simplifying the equation, we get:<br /><br />$x = (x+6) + 10\sqrt{x+6} + 25$<br /><br />Now, let's isolate the square root term:<br /><br />$x - x - 6 = 10\sqrt{x+6} + 25 - x$<br /><br />Simplifying further, we have:<br /><br />$-6 = 10\sqrt{x+6} + 25 - x$<br /><br />Now, let's isolate the square root term again:<br /><br />$-6 - 25 = 10\sqrt{x+6} - x$<br /><br />Simplifying, we get:<br /><br />$-31 = 10\sqrt{x+6} - x$<br /><br />Now, let's isolate the square root term one more time:<br /><br />$-31 + x = 10\sqrt{x+6}$<br /><br />Simplifying, we have:<br /><br />$x - 31 = 10\sqrt{x+6}$<br /><br />Now, let's square both sides of the equation again to eliminate the square root:<br /><br />$(x - 31)^2 = (10\sqrt{x+6})^2$<br /><br />Simplifying, we get:<br /><br />$x^2 - 62x + 961 = 100(x+6)$<br /><br />Now, let's simplify the equation further:<br /><br />$x^2 - 62x + 961 = 100x + 600$<br /><br />Rearranging the terms, we have:<br /><br />$x^2 - 162x + 361 = 0$<br /><br />This is a quadratic equation, which can be solved using the quadratic formula:<br /><br />$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$<br /><br />In this case, $a = 1$, $b = -162$, and $c = 361$. Plugging these values into the quadratic formula, we get:<br /><br />$x = \frac{162 \pm \sqrt{(-162)^2 - 4(1)(361)}}{2(1)}$<br /><br />Simplifying, we have:<br /><br />$x = \frac{162 \pm \sqrt{26244 - 1444}}{2}$<br /><br />$x = \frac{162 \pm \sqrt{24700}}{2}$<br /><br />$x = \frac{162 \pm 157.24}{2}$<br /><br />Therefore, the solutions for $x$ are:<br /><br />$x_1 = \frac{162 + 157.24}{2} = 159.62$<br /><br />$x_2 = \frac{162 - 157.24}{2} = 2.38$<br /><br />So, the correct answer is (A) Additional Solution (-) No Solution.