To solve the equation x² + 8x + 3 = 0 by completing the square, we first need to rewrite the equation in a more manageable form.
1. Start with the original equation:
x² + 8x + 3 = 0
2. To complete the square, we want to focus on the quadratic and linear terms, x² + 8x. Take the coefficient of x, which is 8, divide it by 2 to get 4, and then square it (4² = 16).
3. Now, we’ll rewrite the equation by adding and subtracting this square:
x² + 8x + 16 – 16 + 3 = 0
4. This simplifies to:
(x + 4)² – 13 = 0
5. Now, isolate the squared term:
(x + 4)² = 13
6. Next, take the square root of both sides:
x + 4 = ±√13
7. Finally, solve for x:
x = -4 ± √13
This gives us two solutions:
x = -4 + √13 and x = -4 – √13.
8. Thus, the solution set of the equation is:
{ -4 + √13, -4 – √13 }
Completing the square is a useful method not only for solving quadratic equations but also for understanding the properties of parabolas.