Solve Radical Equations
Solve the radical equation, and check all proposed solutions.
-$$\sqrt { x + 6 } + \sqrt { 2 - x } = 4$$
A) $\{ - 2 \}$
B) $\{ 2 , - 2 \}$
C) $\{ 0 \}$
D) $\{ \sqrt { 31 } , - 2 \}$

Question

Quizplus · Accepted Answer

To solve the equation $\sqrt{x + 6} + \sqrt{2 - x} = 4$, we first isolate one of the square roots and then square both sides of the equation to eliminate the square root. Let's isolate $\sqrt{x + 6}$:$$\sqrt{x + 6} = 4 - \sqrt{2 - x}$$Squaring both sides:$$x + 6 = 16 - 8\sqrt{2 - x} + (2 - x)$$Simplifying:$$x + 6 = 18 - 8\sqrt{2 - x} - x$$$$2x + 8\sqrt{2 - x} = 12$$$$x + 4\sqrt{2 - x} = 6$$Let's isolate the square root again:$$4\sqrt{2 - x} = 6 - x$$Squaring both sides again:$$16(2 - x) = 36 - 12x + x^2$$$$32 - 16x = 36 - 12x + x^2$$Rearranging:$$x^2 + 4x + 4 = 0$$This simplifies to:$$(x + 2)^2 = 0$$Thus, $x = -2$.Checking the solution in the original equation:$$\sqrt{-2 + 6} + \sqrt{2 - (-2)} = \sqrt{4} + \sqrt{4} = 2 + 2 = 4$$The solution $x = -2$ satisfies the original equation, so the correct answer is $\{-2\}$.

Solve Radical Equations Solve the Radical Equation, and Check All x+6+2−x=4\sqrt { x + 6 } + \sqrt { 2 - x } = 4x+6​+2−x​=4

Solve Radical Equations
Solve the Radical Equation, and Check All $\sqrt { x + 6 } + \sqrt { 2 - x } = 4$