solve the following equation by method of completing the perfect square:-
4x^2 + 4 √3 x +3=0
4x^2 + 4√3 + √3 = -3 + √3
(2x + √3)^2 = -3+√3 , the right side is negative, so there will be no real roots
2x + √3 = ±√(3 - √3) i
2x = -√3 ± √(3 - √3) i
x = ( -√3 ± √(3-√3) i )/2
Just realized that the expression already is a perfect square
(2x + √3)^2 = 4x^2 + 4√3x + 3
so (2x+√3)^2 = 0
2x + √3 = 0
2x = -√3
x = - √3/2
To solve the given equation using the method of completing the square, follow these steps:
Step 1: Group the terms with x together:
4x^2 + 4 √3 x + 3 = 0
Step 2: Divide the equation by the coefficient of x^2 (if it's not already 1) to simplify the equation. In this case, divide the entire equation by 4:
x^2 + (√3)x + 3/4 = 0
Step 3: Move the constant term (3/4) to the other side of the equation:
x^2 + (√3)x = -3/4
Step 4: The next step is to take half of the coefficient of x and then square it. In this case, take half of (√3) and square it:
(√3/2)^2 = (3/2)^2 = 9/4
Step 5: Add the value from step 4 to both sides of the equation:
x^2 + (√3)x + 9/4 = -3/4 + 9/4
x^2 + (√3)x + 9/4 = 6/4
x^2 + (√3)x + 9/4 = 3/2
Step 6: Rewrite the left side of the equation as a perfect square:
(x + (√3)/2)^2 = 3/2
Step 7: Take the square root of both sides:
√(x + (√3)/2)^2 = ±√(3/2)
x + (√3)/2 = ±√(3/2)
Step 8: Solve for x by subtracting (√3)/2 from both sides:
x = - (√3)/2 ± √(3/2)
Therefore, the solutions to the given equation 4x^2 + 4 √3 x + 3 = 0 are:
x = - (√3)/2 + √(3/2)
x = - (√3)/2 - √(3/2)