Question

How do you factor the expression ${x^2} + 12x + 20$?

Answer 1

Hint: Factorizing reduces the higher degree equation into its linear equation. In the above given question, we need to reduce the quadratic equation into its simplest form in such a way that addition of products of the factors of first and last term should be equal to the middle term i.e. $12x$.

Complete step by step answer:
$a{x^2} + bx + c$ is a general way of writing quadratic equations where a, b and c are numbers.
In the above expression,
a=1, b=12, c=20
${x^2} + 12x + 20$
First step is by multiplying the coefficient of ${x^2}$ and the constant term 20, we get $20{x^2}$. After this, factors of $20{x^2}$ should be calculated in such a way that their addition should be equal to 12x.
Factors of 20 can be 10 and 2 or 5 and 4. But $5 + 4 \ne 12$, so we will use 10 and 2.
where $10{x^{}} + 2x = 12x$.
So, further we write the equation by equating it with zero and splitting the middle term according to the factors.
$
   \Rightarrow {x^2} + 12x + 20 = 0 \\
   \Rightarrow {x^2} + 10x + 2x + 20 = 0 \\
 $
Now, by grouping the first two and last two terms we get common factors.
$
   \Rightarrow x(x + 10) + 2(x + 10) = 0 \\
   \Rightarrow (x + 2)(x + 10) = 0 \\
 $
Taking x common from the first group and 2 common from the second we get the above equation.
Therefore, by solving the above quadratic equation we get factors -2 and -10.

Note: In quadratic equation, an alternative way of finding the factors is by directly solving the equation by using a formula which is given below:
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
By substituting the values of a=1, b=12 and c=20 we get the factors of x.
$
  x = \dfrac{{ - 12 \pm \sqrt {{{(12)}^2} - 4(1)(20)} }}{{2(1)}} \\
    \\
 $
$x = $-2 or $x = $-10