To solve the equation 3x² + 5x + 2 = 0 by completing the square, we start by rearranging the equation in a form that makes it easier to complete the square.
First, we can factor out the coefficient of x² from the first two terms:
Step 1: Factor out 3:
3(x² + (5/3)x) + 2 = 0
Next, we want to complete the square inside the parentheses.
Step 2: Take half of the coefficient of x (which is 5/3), square it, and add and subtract it inside the parentheses:
3(x² + (5/3)x + (25/36) - (25/36)) + 2 = 0
This simplifies to:
3((x + 5/6)² - (25/36)) + 2 = 0
Step 3: Distribute the 3:
3(x + 5/6)² - 75/36 + 2 = 0
Convert 2 into a fraction with a denominator of 36:
3(x + 5/6)² - 75/36 + 72/36 = 0
Now combining the constants gives:
3(x + 5/6)² - 3/36 = 0
Step 4: Add 3/36 to both sides:
3(x + 5/6)² = 3/36
Divide both sides by 3:
(x + 5/6)² = 1/36
Step 5: Take the square root of both sides:
x + 5/6 = ±1/6
Step 6: Solve for x:
x = -5/6 ± 1/6
This gives us two solutions:
x = -4/6 = -2/3
x = -6/6 = -1
Final Answer: The solutions to the equation 3x² + 5x + 2 = 0 are x = -2/3 and x = -1.