Solve Radical Equations
Solve the radical equation, and check all proposed solutions.
-$$\sqrt { 30 x + 15 } = x + 8$$
A) $\{ 7 \}$
B) $\{ - 6 \}$
C) $\{ - 7 \}$
D) $\{ 8 \}$

Question

Quizplus · Accepted Answer

We square both sides of the equation to eliminate the square root:
\begin{align*}
\sqrt{30x+15}&=x+8\
\Rightarrow\qquad (\sqrt{30x+15})^2&=(x+8)^2\
\Rightarrow\qquad 30x+15&=x^2+16x+64\
\Rightarrow\qquad x^2+16x+49&=0
\end{align*}
We can factor the left side of the equation as $(x+7)^2=0$. Therefore, the only solution is $x=-7$.
However, we need to check whether $-7$ is a valid solution by substituting it back into the original equation:
$$\sqrt{30(-7)+15}=-7+8$$
$$\sqrt{-195}=1$$
The left side of the equation is undefined for real numbers, which means $-7$ is not a valid solution.
Therefore, the only solution to the original equation is $x=7$, and we can check it by substituting it back into the original equation:
$$\sqrt{30(7)+15}=7+8$$
$$6\sqrt{10}=15$$
The equation is true, so $x=7$ is a valid solution.

Solve Radical Equations Solve the Radical Equation, and Check All 30x+15=x+8\sqrt { 30 x + 15 } = x + 830x+15​=x+8

Solve Radical Equations
Solve the Radical Equation, and Check All $\sqrt { 30 x + 15 } = x + 8$