Question Solve for all values of x: sqrt (x)-5=sqrt (x+10) Answer Attempt 1out of 2 x=

To solve the equation $\sqrt{x} - 5 = \sqrt{x+10}$, 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 />$\sqrt{x} = \sqrt{x+10} + 5$<br /><br />Next, we can square both sides of the equation to eliminate the square roots:<br /><br />$(\sqrt{x})^2 = (\sqrt{x+10} + 5)^2$<br /><br />Simplifying the right side of the equation, we get:<br /><br />$x = (x+10) + 10\sqrt{x+10} + 25$<br /><br />Now, let's isolate the square root term by subtracting $x+10$ from both sides:<br /><br />$x - (x+10) = 10\sqrt{x+10} + 25 - (x+10)$<br /><br />Simplifying further, we have:<br /><br />$-10 = 10\sqrt{x+10} + 15 - x$<br /><br />Next, let's isolate the square root term by subtracting 15 from both sides:<br /><br />$-10 - 15 = 10\sqrt{x+10} - x$<br /><br />Simplifying, we get:<br /><br />$-25 = 10\sqrt{x+10} - x$<br /><br />Now, let's isolate the square root term by adding $x$ to both sides:<br /><br />$-25 + x = 10\sqrt{x+10}$<br /><br />Finally, let's isolate the square root term by dividing both sides by 10:<br /><br />$\frac{-25 + x}{10} = \sqrt{x+10}$<br /><br />To solve for $x$, we can square both sides of the equation again:<br /><br />$\left(\frac{-25 + x}{10}\right)^2 = (\sqrt{x+10})^2$<br /><br />Simplifying, we get:<br /><br />$\frac{625 - 50x + x^2}{100} = x + 10$<br /><br />Multiplying both sides by 100, we have:<br /><br />$625 - 50x + x^2 = 100x + 1000$<br /><br />Rearranging the terms, we get a quadratic equation:<br /><br />$x^2 - 150x + 375 = 0$<br /><br />We can solve this quadratic equation using the quadratic formula:<br /><br />$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$<br /><br />In this case, $a = 1$, $b = -150$, and $c = 375$. Plugging these values into the quadratic formula, we get:<br /><br />$x = \frac{150 \pm \sqrt{150^2 - 4(1)(375)}}{2(1)}$<br /><br />Simplifying further, we have:<br /><br />$x = \frac{150 \pm \sqrt{22500 - 1500}}{2}$<br /><br />$x = \frac{150 \pm \sqrt{21000}}{2}$<br /><br />$x = \frac{150 \pm 100\sqrt{21}}{2}$<br /><br />Therefore, the solutions for $x$ are:<br /><br />$x = \frac{150 + 100\sqrt{21}}{2}$ or $x = \frac{150 - 100\sqrt{21}}{2}$<br /><br />So, the final answer is:<br /><br />$x = \frac{150 + 100\sqrt{21}}{2}$ or $x = \frac{150 - 100\sqrt{21}}{2}$