Solve Radical Equations
-$$\sqrt { 2 x + 5 } - \sqrt { x - 2 } = 3$$
A) $\{ 2,38 \}$
B) $\{ 3,8 \}$
C) $\{ 2 \}$
D) $\{ - 2 \}$

Question

Quizplus · Accepted Answer

To solve the equation $\sqrt{2x + 5} - \sqrt{x - 2} = 3$, first isolate one of the square roots:$\sqrt{2x + 5} = \sqrt{x - 2} + 3$Square both sides to eliminate the square root:$(\sqrt{2x + 5})^2 = (\sqrt{x - 2} + 3)^2$$2x + 5 = (x - 2) + 6\sqrt{x - 2} + 9$Simplify:$2x + 5 = x + 7 + 6\sqrt{x - 2}$$x - 2 = 6\sqrt{x - 2}$Square both sides again to eliminate the square root:$(x - 2)^2 = 36(x - 2)$Expand and simplify:$x^2 - 4x + 4 = 36x - 72$$x^2 - 40x + 76 = 0$Solve the quadratic equation, which can be factored or solved using the quadratic formula. The solutions are $x = 2$ and $x = 38$. However, it's important to check both solutions in the original equation because squaring both sides can introduce extraneous solutions.For $x = 2$, the original equation becomes $\sqrt{2(2) + 5} - \sqrt{2 - 2} = 3$, which simplifies to $\sqrt{9} - \sqrt{0} = 3$, or $3 = 3$, which is true.For $x = 38$, the original equation becomes $\sqrt{2(38) + 5} - \sqrt{38 - 2} = 3$, which simplifies to $\sqrt{81} - \sqrt{36} = 3$, or $9 - 6 = 3$, which is also true.Therefore, both solutions are valid, and the correct answer is $\{2, 38\}$.

Solve Radical Equations - 2x+5−x−2=3\sqrt { 2 x + 5 } - \sqrt { x - 2 } = 32x+5​−x−2​=3

Solve Radical Equations
- $\sqrt { 2 x + 5 } - \sqrt { x - 2 } = 3$