Factoring a quadratic equation like x² + 5x + 6 involves finding two binomials that multiply to give the original expression. Here’s a step-by-step guide:
- Identify the coefficients: In this equation, the coefficients are:
– a (coefficient of x²) = 1
– b (coefficient of x) = 5
– c (constant term) = 6
Our goal is to find two numbers that:
- Add up to b (which is 5).
- Multiply to a * c (which is 1 * 6 = 6).
Now, let’s think of pairs of factors of 6:
- 1 and 6
(1 + 6 = 7, does not work) - 2 and 3
(2 + 3 = 5, works!)
Having found the two numbers, we can express the quadratic in factored form:
This gives us:
- (x + 2)(x + 3)
To verify, you can expand the factors:
(x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6
This confirms our factorization is accurate!
Thus, the complete factorization of x² + 5x + 6 is:
- (x + 2)(x + 3)
Factoring helps in finding the roots of the equation, which can be set to zero:
(x + 2)(x + 3) = 0
- So, x + 2 = 0 gives x = -2
- And x + 3 = 0 gives x = -3
In conclusion, factoring not only helps simplify the expression but also aids in solving equations. Happy learning!