**Solving by Completing the Square Method**

To solve the equation √3x^2 + 10x + 7√3, we can use the completing the square method. This method involves transforming the quadratic equation into a perfect square trinomial. Here's how you can do it step by step:

**Step 1: Divide by the coefficient of x^2**
To start, divide the entire equation by the coefficient of x^2, which in this case is √3. This will simplify the equation and make it easier to work with. After dividing, the equation becomes:

x^2 + (10/√3)x + (7√3/√3)

Simplifying the last term, we get:

x^2 + (10/√3)x + 7√3

**Step 2: Move the constant term to the other side**
Next, move the constant term (7√3) to the right side of the equation. This can be done by subtracting it from both sides:

x^2 + (10/√3)x = -7√3

**Step 3: Complete the square on the left side**
To complete the square on the left side of the equation, we need to add a certain value to both sides. The value we add is half the coefficient of x, squared. In this case, the coefficient of x is (10/√3), so half of it is (5/√3). Squaring this gives us 25/3.

Add 25/3 to both sides of the equation:

x^2 + (10/√3)x + 25/3 = -7√3 + 25/3

**Step 4: Write the left side as a perfect square trinomial**
The left side of the equation can now be written as a perfect square trinomial. To do this, factor the left side:

(x + (5/√3))^2 = -7√3 + 25/3

**Step 5: Simplify the right side**
Simplify the right side of the equation by finding a common denominator and combining the terms:

(x + (5/√3))^2 = (-21√3 + 25)/3

**Step 6: Solve for x**
To solve for x, we need to take the square root of both sides. Remember to consider both the positive and negative square roots:

x + (5/√3) = ±√((-21√3 + 25)/3)

Subtract (5/√3) from both sides:

x = - (5/√3) ±√((-21√3 + 25)/3)

Thus, the solution to the equation √3x^2 + 10x + 7√3 by completing the square method is x = - (5/√3) ±√((-21√3 + 25)/3).

√3x2 + 10x + 7√3 = 0x2 + 10 /√3x + 7 = 0x2 + 10 /√3x=-7Third term = ( 1/2 * coefficient of x)2= ( 1/2 * 10 /√3 )2= ( 5 /√3 )2= 25 / 3Therefore,x2+ 10 /√3x + 25/3=-7 + 25 / 3 ... (adding to both sides)x2+ 10 /√3x + 25/3 = 4 / 3( x - 5 /√3 ) =±2 /√3 ... (taking square roots)Hence,x = 7 /√3 ,√3