To solve the quadratic equation x² + 7x + 12 = 0, we can use the factoring method or the quadratic formula. Here, I’ll demonstrate both methods for clarity.
Method 1: Factoring
1. First, we need to find two numbers that multiply to the constant term (12) and add up to the coefficient of the x term (7).
2. The numbers 3 and 4 meet these criteria:
- 3 × 4 = 12
- 3 + 4 = 7
3. Using these numbers, we can factor the quadratic:
(x + 3)(x + 4) = 0
4. Now, we can set each factor equal to zero:
- x + 3 = 0
- x + 4 = 0
5. Solving these equations gives:
- x = -3
- x = -4
So the solutions to the equation are x = -3 and x = -4.
Method 2: Quadratic Formula
If factoring doesn’t seem straightforward, we can always use the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a
In our equation, a = 1, b = 7, and c = 12.
1. Calculate the discriminant:
- b² – 4ac = 7² – 4(1)(12)
- = 49 – 48 = 1
2. Now substitute into the quadratic formula:
x = (–7 ± √1) / 2
x = (–7 ± 1) / 2
3. This gives us two solutions:
- x = (–7 + 1) / 2 = –6 / 2 = –3
- x = (–7 – 1) / 2 = –8 / 2 = –4
Again, we arrive at the same solutions: x = -3 and x = -4.
In conclusion, whether you factor the quadratic equation or use the quadratic formula, the solutions for the equation x² + 7x + 12 = 0 are x = -3 and x = -4.