x^2+5x+6
The Correct Answer and Explanation is :
To solve the quadratic equation ( x^2 + 5x + 6 = 0 ), we can use factoring.
Step 1: Look for two numbers that multiply to 6 and add up to 5.
The equation is in the standard form of ( ax^2 + bx + c = 0 ), where ( a = 1 ), ( b = 5 ), and ( c = 6 ). We need to find two numbers that:
- Multiply to give ( c ) (in this case, 6), and
- Add up to give ( b ) (in this case, 5).
The pair of numbers that work here are 2 and 3:
- ( 2 \times 3 = 6 ),
- ( 2 + 3 = 5 ).
Step 2: Factor the quadratic expression.
We can now express the quadratic equation as a product of two binomials. The equation ( x^2 + 5x + 6 ) factors as:
[
(x + 2)(x + 3) = 0
]
Step 3: Solve for ( x ).
To solve for ( x ), set each factor equal to 0:
[
x + 2 = 0 \quad \text{or} \quad x + 3 = 0
]
Solving each equation:
- ( x + 2 = 0 ) gives ( x = -2 ),
- ( x + 3 = 0 ) gives ( x = -3 ).
Thus, the solutions to the quadratic equation ( x^2 + 5x + 6 = 0 ) are ( x = -2 ) and ( x = -3 ).
Explanation of Factoring:
Factoring is a method of solving quadratic equations by rewriting the quadratic expression as a product of two binomials. This method works well when the quadratic equation can be factored easily, which happens when you can find two numbers that multiply to the constant term and add up to the coefficient of the middle term.
In this case, we found that ( 2 \times 3 = 6 ) and ( 2 + 3 = 5 ), which allowed us to factor the quadratic expression as ( (x + 2)(x + 3) = 0 ). Setting each factor equal to 0 gives the solutions ( x = -2 ) and ( x = -3 ). Factoring is a fast and efficient method for solving many quadratic equations when the factoring is straightforward.