Solve the radical equation, and check all proposed solutions.

Question

Solve the radical equation, and check all proposed solutions. $$ \sqrt{4 x+12}=x $$

UpStudy AI · Accepted Answer

Solve the equation $$ \sqrt{4x+12}=x $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\sqrt{4x+12}=x$$
- step1: Find the domain:
  $$\sqrt{4x+12}=x ,x\geq -3$$
- step2: Simplify the root:
  $$2\sqrt{x+3}=x$$
- step3: Rewrite the expression:
  $$\sqrt{x+3}=\frac{x}{2}$$
- step4: Evaluate:
  $$\sqrt{x+3}=\frac{x}{2},\frac{x}{2}\geq 0$$
- step5: Evaluate:
  $$\sqrt{x+3}=\frac{x}{2},x\geq 0$$
- step6: Solve the equation:
  $$\begin{align}&x=6\&x=-2\end{align},x\geq 0$$
- step7: Find the intersection:
  $$x=6$$
- step8: Check if the solution is in the defined range:
  $$x=6,x\geq -3$$
- step9: Find the intersection:
  $$x=6$$
- step10: Check the solution:
  $$x=6$$
The solution to the radical equation $$ \sqrt{4x+12}=x $$ is $$ x=6 $$.

Now, we need to check if this solution satisfies the original equation. Substituting $$ x=6 $$ into the original equation, we have:

$$
\sqrt{4(6)+12} = 6
$$

Simplifying the expression inside the square root:

$$
\sqrt{24+12} = 6
$$

$$
\sqrt{36} = 6
$$

Since $$ \sqrt{36} = 6 $$, the solution $$ x=6 $$ satisfies the original equation. Therefore, the solution to the radical equation $$ \sqrt{4x+12}=x $$ is $$ x=6 $$.
$$ x=6 $$ is the solution to the equation $$ \sqrt{4x+12}=x $$.

Solve the radical equation, and check all proposed solutions.

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