Answer

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**Hint:**In this question to write the factor of ${w^2} + 18w + 77$ expression by converting it into a quadratic equation then find its roots to write it in factored form. We can find the factor of this quadratic equation by the factor method of finding roots for a quadratic equation. But we will find the root of the quadratic equation by quadratic formula.

**Complete step by step solution:**

Let us try to solve this question in which we need the factor of a given quadratic equation ${w^2} + 18w + 77 = 0$. To find the factor of this quadratic, we will first find the discriminant of this quadratic equation and from which we find the nature of the root of this quadratic equation. After which we will find the root of the quadratic equation using quadratic formula and factor it.

Types of root of quadratic equation: $a{w^2} + bw + c = 0$

1. Two distinct real roots, if ${b^2} - 4ac > 0$( which is called discriminant of this quadratic equation)

2. Two equal real roots, if ${b^2} - 4ac = 0$

3. No real roots if,${b^2} - 4ac < 0$

In this quadratic equation${w^2} + 18w + 77 = 0$, we have

$

a = 1 \\

b = 18 \\

c = 77 \\

$

So the discriminant of this quadratic equation is

$

{b^2} - 4ac = \,{(18)^2} - 4\times 1\times 77 \\

= \,324 - 308 \\

= 16 > 0 \\

$

Hence this equation has two distinct equal roots.

So the root of this equation by using quadratic formula are,

$

w = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} = \,\dfrac{{ - 18 \pm \sqrt {{{18}^2} - 4\times 1\times 77}}}{{2\times 1}} \\

w = \,\dfrac{{ - 18 \pm \sqrt {324 - 308} }}{2} \\

w= \,\dfrac{{ - 18 \pm \sqrt {16} }}{2} \\

$

As we know that $\sqrt {16} = \pm 4$ we get

$w = \,\dfrac{{ - 18 \pm 4}}{2}$

$w = - 9 \pm 2$ (Dividing by$2$)

So the roots of the given equation are $w = - 11$ and $w = - 7$.

**Hence the expression ${w^2} + 18w + 77$ in factored form will be written as $(w + 11)(w + 7)$.**

**Note:**While solving these types of questions we will first change the given expression into a quadratic equation and then check the nature of the root and then find the root of that quadratic equation using quadratic formula. To solve this type of question you need to know the conditions for the nature of roots.

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