Quadratic equations are those equations which have a degree of 2. These equations can be written in the form of ax?? + bx + c = 0, where a, b, and c are constants. When a is not equal to 1, then the quadratic equation is called a non-monic quadratic equation. In this article, we will discuss how to factorise non-monic quadratics in a step-by-step manner.
Step 1: Make the Coefficient of x?? equal to 1
The first step in factorising a non-monic quadratic equation is to make the coefficient of x?? equal to 1. To do this, we need to divide the entire equation by the coefficient of x??. For example, consider the equation 3x?? + 5x + 2 = 0. We divide the equation by 3 to get x?? + (5/3)x + (2/3) = 0.
Step 2: Find the Factors of the Constant Term
The second step in factorising a non-monic quadratic equation is to find the factors of the constant term. The constant term is the term which does not have an x in it. For example, in the equation x?? + (5/3)x + (2/3) = 0, the constant term is (2/3). We need to find two numbers which multiply to give (2/3) and add up to the coefficient of x, which is (5/3). In this case, the factors are (1/3) and (2/3).
Step 3: Rewrite the Middle Term as the Sum of the Two Factors
The third step in factorising a non-monic quadratic equation is to rewrite the middle term as the sum of the two factors we found in step 2. For example, in the equation x?? + (5/3)x + (2/3) = 0, we rewrite (5/3)x as (1/3)x + (2/3)x. This gives us the equation x?? + (1/3)x + (2/3)x + (2/3) = 0.
Step 4: Group the First Two Terms and the Last Two Terms Separately
The fourth step in factorising a non-monic quadratic equation is to group the first two terms and the last two terms separately. For example, in the equation x?? + (1/3)x + (2/3)x + (2/3) = 0, we group the first two terms as x(x + 1/3) and the last two terms as (2/3)(x + 1/3). This gives us the equation x(x + 1/3) + (2/3)(x + 1/3) = 0.
Step 5: Factorise the Common Terms
The fifth and final step in factorising a non-monic quadratic equation is to factorise the common terms. For example, in the equation x(x + 1/3) + (2/3)(x + 1/3) = 0, we can factorise out (x + 1/3) to get (x + 1/3)(x + 2/3) = 0. This gives us the two solutions x = -1/3 and x = -2/3.
Conclusion
In conclusion, factorising non-monic quadratic equations can be done in five easy steps. These steps involve making the coefficient of x?? equal to 1, finding the factors of the constant term, rewriting the middle term as the sum of the two factors, grouping the first two terms and the last two terms separately, and factorising the common terms. By following these steps, you can easily factorise non-monic quadratic equations and solve for the values of x.
