To solve the equation [tex]\(4 \sqrt{6x} = \sqrt{4x^2}\)[/tex] for [tex]\(x\)[/tex], we will follow these steps:
1. Simplify Both Sides of the Equation:
First, simplify the square root terms. The equation given is:
[tex]\[ 4 \sqrt{6x} = \sqrt{4x^2} \][/tex]
Simplifying the right-hand side:
[tex]\[ \sqrt{4x^2} = 2x \][/tex]
because the square root of [tex]\(4x^2\)[/tex] is [tex]\(2x\)[/tex].
So, the equation now becomes:
[tex]\[ 4 \sqrt{6x} = 2x \][/tex]
2. Isolate the Square Root Term:
Next, we isolate the square root term by dividing both sides of the equation by 4:
[tex]\[ \sqrt{6x} = \frac{2x}{4} \][/tex]
[tex]\[ \sqrt{6x} = \frac{x}{2} \][/tex]
3. Square Both Sides:
To eliminate the square root, we square both sides of the equation:
[tex]\[ (\sqrt{6x})^2 = \left(\frac{x}{2}\right)^2 \][/tex]
[tex]\[ 6x = \frac{x^2}{4} \][/tex]
4. Solve the Quadratic Equation:
Now, we multiply both sides by 4 to clear the fraction:
[tex]\[ 4 \cdot 6x = x^2 \][/tex]
[tex]\[ 24x = x^2 \][/tex]
Rearrange this to form a standard quadratic equation:
[tex]\[ x^2 - 24x = 0 \][/tex]
Factor the quadratic:
[tex]\[ x(x - 24) = 0 \][/tex]
So, the solutions to this equation are:
[tex]\[ x = 0 \quad \text{or} \quad x = 24 \][/tex]
5. Check Possible Values Against Given Choices:
Finally, we compare our solutions with the provided options. The possible values of [tex]\(x\)[/tex] are 0 and 24. However, 0 is not a choice in the given options.
The correct and possible value of [tex]\(x\)[/tex] among the given choices is:
[tex]\[ \boxed{24} \][/tex]
