Find all real solutions of the equation. $$\sqrt { 5 x + 9 } + 3 = x$$
A) 0
B) 0,11
C) 11
D) 1,0
E) $- 11$

Question

Quizplus · Accepted Answer

To solve the equation $\sqrt{5x + 9} + 3 = x$, we first isolate the square root by subtracting 3 from both sides, giving us $\sqrt{5x + 9} = x - 3$. Squaring both sides to eliminate the square root gives us $5x + 9 = (x - 3)^2$. Expanding the right side yields $5x + 9 = x^2 - 6x + 9$. Rearranging the equation to set it to 0 gives us $x^2 - 11x = 0$, which simplifies to $x(x - 11) = 0$. This gives us two possible solutions for $x$: $x = 0$ and $x = 11$. However, we must check these solutions in the original equation to ensure they don't result from squaring both sides (which can introduce extraneous solutions).For $x = 0$, the original equation becomes $\sqrt{5(0) + 9} + 3 = 0$, which simplifies to $\sqrt{9} + 3 = 0$, or $3 + 3 = 0$, which is not true. Therefore, $x = 0$ is not a solution.For $x = 11$, the original equation becomes $\sqrt{5(11) + 9} + 3 = 11$, which simplifies to $\sqrt{55 + 9} + 3 = 11$, or $\sqrt{64} + 3 = 11$, which simplifies to $8 + 3 = 11$, which is true. Therefore, $x = 11$ is a valid solution.Thus, the only real solution to the equation is $x = 11$.

Find All Real Solutions of the Equation 5x+9+3=x\sqrt { 5 x + 9 } + 3 = x5x+9​+3=x

Find All Real Solutions of the Equation $\sqrt { 5 x + 9 } + 3 = x$