Solve for all values of sqrt (x)+4=sqrt (2x+7) Answer Attempt 1 out of 2 Additional Solution No Solution z=

To solve the equation $\sqrt{x} + 4 = \sqrt{2x + 7}$, we can follow these steps:<br /><br />1. Isolate one of the square roots on one side of the equation:<br /> $\sqrt{x} + 4 = \sqrt{2x + 7}$<br /> $\sqrt{x} = \sqrt{2x + 7} - 4$<br /><br />2. Square both sides of the equation to eliminate the square root:<br /> $(\sqrt{x})^2 = (\sqrt{2x + 7} - 4)^2$<br /> $x = (2x + 7) - 8\sqrt{2x + 7} + 16$<br /> $x = 2x + 7 - 8\sqrt{2x + 7} + 16$<br /> $x = 2x + 23 - 8\sqrt{2x + 7}$<br /><br />3. Isolate the square root term:<br /> $x - 2x = 23 - 8\sqrt{2x + 7}$<br /> $-x = 23 - 8\sqrt{2x + 7}$<br /> $x = 8\sqrt{2x + 7} - 23$<br /><br />4. Square both sides of the equation again to eliminate the square root:<br /> $x^2 = (8\sqrt{2x + 7} - 23)^2$<br /> $x^2 = 64(2x + 7) - 736\sqrt{2x + 7} + 529$<br /> $x^2 = 128x + 448 - 736\sqrt{2x + 7} + 529$<br /> $x^2 = 128x + 977 - 736\sqrt{2x + 7}$<br /><br />5. Isolate the square root term:<br /> $x^2 - 128x - 977 = -736\sqrt{2x + 7}$<br /> $x^2 - 128x - 977 = -736\sqrt{2x + 7}$<br /> $x^2 - 128x - 977 + 736\sqrt{2x + 7} = 0$<br /><br />6. Solve the resulting quadratic equation:<br /> $x^2 - 128x - 977 + 736\sqrt{2x + 7} = 0$<br /> $x = \frac{128 \pm \sqrt{128^2 - 4(1)(-977 + 736\sqrt{2x + 7})}}{2(1)}$<br /> $x = \frac{128 \pm \sqrt{16384 - 4(-977 + 736\sqrt{2x + 7})}}{2}$<br /> $x = \frac{128 \pm \sqrt{16384 + 3908 - 2944\sqrt{2x + 7}}}{2}$<br /> $x = \frac{128 \pm \sqrt{20292 - 2944\sqrt{2x + 7}}}{2}$<br /><br />The solution to the equation is:<br />$x = \frac{128 \pm \sqrt{20292 - 2944\sqrt{2x + 7}}}{2}$<br /><br />Note: The solution involves a square root term, which means it may have multiple solutions depending on the value of $x$.