Questions: Solve for (x). [ 3 sqrtx+2=sqrtx+4 ] (x=5) (x=-frac74) (x=-1) (x=frac112)

Solution Steps

We start with the equation: \[ 3 \sqrt{x + 2} = \sqrt{x + 4} \]

To eliminate the square roots, we square both sides of the equation: \[ (3 \sqrt{x + 2})^2 = (\sqrt{x + 4})^2 \] This simplifies to: \[ 9(x + 2) = x + 4 \]

Next, we rearrange the equation to isolate terms involving \(x\): \[ 9x + 18 = x + 4 \] Subtract \(x\) from both sides: \[ 8x + 18 = 4 \] Then, subtract 18 from both sides: \[ 8x = 4 - 18 \] This simplifies to: \[ 8x = -14 \]

Now, we divide both sides by 8 to solve for \(x\): \[ x = -\frac{14}{8} = -\frac{7}{4} \]

Finally, we check if \(x = -\frac{7}{4}\) satisfies the original equation: Substituting \(x = -\frac{7}{4}\) into the left side: \[ 3 \sqrt{-\frac{7}{4} + 2} = 3 \sqrt{-\frac{7}{4} + \frac{8}{4}} = 3 \sqrt{\frac{1}{4}} = 3 \cdot \frac{1}{2} = \frac{3}{2} \] Now for the right side: \[ \sqrt{-\frac{7}{4} + 4} = \sqrt{-\frac{7}{4} + \frac{16}{4}} = \sqrt{\frac{9}{4}} = \frac{3}{2} \] Since both sides are equal, \(x = -\frac{7}{4}\) is indeed a valid solution.

Final Answer