Answer :
To solve the quadratic equation [tex]\( x^2 + 7x + 12 = 0 \)[/tex], we can use the factoring method. Here is the detailed, step-by-step solution:
### B. Factoring
1. Rewrite the equation:
[tex]\[ x^2 + 7x + 12 = 0 \][/tex]
2. Identify the constant term and the coefficient of the middle term:
- The constant term is 12.
- The coefficient of the middle term (x) is 7.
3. Find two numbers that multiply to the constant term (12) and add up to the coefficient of the middle term (7):
- The numbers are 3 and 4 because [tex]\( 3 \times 4 = 12 \)[/tex] and [tex]\( 3 + 4 = 7 \)[/tex].
4. Use these numbers to factor the quadratic:
[tex]\[ x^2 + 7x + 12 = (x + 3)(x + 4) \][/tex]
5. Set each factor to zero and solve for x:
[tex]\[ (x + 3) = 0 \quad \text{or} \quad (x + 4) = 0 \][/tex]
6. Solve each equation:
[tex]\[ x + 3 = 0 \implies x = -3 \][/tex]
[tex]\[ x + 4 = 0 \implies x = -4 \][/tex]
Therefore, the solutions to the equation [tex]\( x^2 + 7x + 12 = 0 \)[/tex] are:
[tex]\[ x = -3 \quad \text{and} \quad x = -4 \][/tex]
### B. Factoring
1. Rewrite the equation:
[tex]\[ x^2 + 7x + 12 = 0 \][/tex]
2. Identify the constant term and the coefficient of the middle term:
- The constant term is 12.
- The coefficient of the middle term (x) is 7.
3. Find two numbers that multiply to the constant term (12) and add up to the coefficient of the middle term (7):
- The numbers are 3 and 4 because [tex]\( 3 \times 4 = 12 \)[/tex] and [tex]\( 3 + 4 = 7 \)[/tex].
4. Use these numbers to factor the quadratic:
[tex]\[ x^2 + 7x + 12 = (x + 3)(x + 4) \][/tex]
5. Set each factor to zero and solve for x:
[tex]\[ (x + 3) = 0 \quad \text{or} \quad (x + 4) = 0 \][/tex]
6. Solve each equation:
[tex]\[ x + 3 = 0 \implies x = -3 \][/tex]
[tex]\[ x + 4 = 0 \implies x = -4 \][/tex]
Therefore, the solutions to the equation [tex]\( x^2 + 7x + 12 = 0 \)[/tex] are:
[tex]\[ x = -3 \quad \text{and} \quad x = -4 \][/tex]
