Question

How do you factor $ {x^2} + 9x + 20 $ ?

Answer 1

Hint: Use the splitting the middle term technique to factorize quadratic equations of the form $ a{x^2} + bx + c $ . In this method, you will need to split the 9x term in such a way that the addition of the coefficients of the splits must be equal to 9 and the product must be equal to 20. After completing this step, you will need to take the common terms out of the equation which will in the end, give you the final answer of the form \[(qx + w)(zx + v)\] , where \[(qx + w)\] and \[(zx + v)\] will be the two factors.

Complete step-by-step answer:
The question has given us \[(ax + b)(cx + d)\]
In order to factorize the question, we will follow the general rules to factorize a quadratic equation using the splitting the middle term technique of the form $ a{x^2} + bx + c $ and that is:-
Step 1: Split the coefficient of x such that addition of those 2 terms equals to $ a $ and product of those terms equals the constant $ c $
Step 2: Bring out the common things out of both the terms and further simplify the equation to get the factors.
So,
If we use trial and error method, 5 and 4 are the two numbers whose additions equals 9 and product equals 20.
So, re writing the equation again, we will get
 \[{x^2} + 9x + 20 = {x^2} + 5x + 4x + 20\]
Taking x common from \[{x^2} + 5x\] and 4 common from \[4x + 20\]
 \[{x^2} + 5x + 4x + 20 = x(x + 5) + 4(x + 5)\]
Now taking \[(x + 5)\] common from both, we will get
 \[x(x + 5) + 4(x + 5) = (x + 5)(x + 4)\]
We can clearly see that the equation has been reduced to a multiplicative form of two terms. Therefore,
 \[(x + 5)\] And \[(x + 4)\] are the two factors of $ {x^2} + 9x + 20 $
So, the correct answer is “ \[(x + 5)\] And \[(x + 4)\] ”.

Note: While splitting the middle term you must remember that the addition of the split must be equal to the coefficient of x and product must be equal to the constant and not the vice versa. If done the other way round, then you will find no common terms in the next steps which will obviously get you to the wrong answer.