Question

How do we factor completely ${x^2} + 11x + 30$ ?

Answer 1

Hint: To solve this question, first we will split the middle term of the given expression by taking two numbers which are multiplied to get $a.c$ and addition to get $b$ . And then conclude the final common factor for the given expression.

Complete step by step answer:
Given expression:
${x^2} + 11x + 30$
To find the factors of the above equation, we can split the Middle Term of the expression to factorise it.
In this method, if we have to factorise an expression like $a{x^2} + bx + c$ , we need to think on two numbers such that:
$ \Rightarrow {N_1}.{N_2} = a.c = 1.30 = 30$
And, ${N_1} + {N_2} = b = 11$
After trying out a few numbers, we get ${N_1} = 5$ and ${N_2} = 6$ :
$5.6 = 30$ , and $5 + 6 = 11$
Now,
$
\Rightarrow {x^2} + 11x + 30 \\
\Rightarrow {x^2} + 5x + 6x + 30 \\
\Rightarrow x(x + 5) + 6(x + 5) \\
\Rightarrow (x + 5)(x + 6) \\
$
Hence, $(x + 6)(x + 5)$ is the factored form of the expression.

Note: The given equation is the quadratic form of expression. In a quadratic expression, the a (the variable raised to the second power) can’t be zero. If a was allowed to be 0, then the x to the power of 2 would be multiplied by zero. It wouldn’t be a quadratic expression anymore. The variables b or c can be 0, but a cannot.