Question

How do you factor the expression ${x^2} + 13x + 36$?

Answer 1

Hint:Factoring 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. $13x$

Complete step by step solution:
$a{x^2} + bx + c$ is a general way of writing quadratic equations where a, b and c are numbers.

In the above expression,
a=1, b=13, c=36
${x^2} + 13x + 36$

First step is by multiplying the coefficient of ${x^2}$ and the constant term 36, we get $36{x^2}$.

After this, factors of $36{x^2}$should be calculated in such a way that their addition should be equal to 13x.

Factors of 36 can be 4 and 9 or 6 and 6. But $6 + 6 \ne 13$,so we will use 4 and 9.
where $4{x^{}} + 9x = 13x$.

So, further we write the equation by equating it with zero and splitting the middle term according to the factors.
$
\Rightarrow {x^2} + 13x + 36 = 0 \\
\Rightarrow {x^2} + 4x + 9x + 36 = 0 \\
$
Now, by grouping the first two and last two terms we get common factors.
$
\Rightarrow x(x + 4) + 9(x + 4) = 0 \\
\Rightarrow (x + 4)(x + 9) = 0 \\
$
Taking x common from the first group and 9 common from the second we get the above equation.

Therefore, by solving the above quadratic equation we get factors -4 and -9.

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=13 and c=36 we get the factors of x.
$
x = \dfrac{{ - 13 \pm \sqrt {{{(13)}^2} - 4(1)(36)} }}{{2(1)}} \\
\\
$
So, the values are $x = $-4 or $x = $-9.