: Factorize the given expression.
\[{{x}^{2}}+7x-18\]

 Hint:Consider 2 numbers a and b. Take the product as (-18) and their sum as 7.Thus split the second term of expression into 2 terms and factor out the common terms.

Complete step-by-step answer:
Let us consider two numbers ‘a’ and ‘b’. The product of a and b should be -18 and their sum should be 7. We have been given the expression \[{{x}^{2}}+7x-18\].
Thus the product ab = -18 and their sum should be 7.
We have been given the expression \[{{x}^{2}}+7x-18\].
Thus the product, ab = -18 and sum, a + b = 7.
To find those numbers, write down the factors of -18, which are:
\[\left[ \left( -1,1 \right),\left( -2,2 \right),\left( -3,3 \right),\left( -6,6 \right),\left( -9,9 \right),\left( -18,18 \right) \right]\]
Now you need to find the value of a and b which makes -18.
Thus,
\[\begin{align}
  & ab=-18 \\
 & \Rightarrow 9\times (-2)=-18 \\
\end{align}\]
Thus, a = 9 and b = -2.
Let’s add a and b, we get,
\[a+b=9+(-2)=7\]
Now let us write down the second term of the expression (7x) as a sum of two terms whose coefficients are the two numbers.
\[{{x}^{2}}+7x-18={{x}^{2}}+9x-2x-18\]
Now let us write the factor out of the first 2 terms and other 2 terms.
\[{{x}^{2}}+9x-2x-18=x(x+9)-2(x+9)=(x-2)(x+9)\]
Thus we got \[{{x}^{2}}+9x-2x-18=(x-2)(x+9)\].
This can’t be factored further. Thus the factorized value of \[{{x}^{2}}+9x-2x-18\] is \[(x-2)(x+9)\].

Note:We can also check these values by using the quadratic formula. The expression \[{{x}^{2}}+9x-2x-18\]is similar to the general expression \[a{{x}^{2}}+bx+c=0\].
Comparing them, we get, a = 1, b = 7 and c = -18.
Substitute these values in \[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\].
\[\Rightarrow \dfrac{-7\pm \sqrt{{{(7)}^{2}}-4\times 1\times (-18)}}{2\times 1}=\dfrac{-7\pm \sqrt{121}}{2}=\dfrac{-7\pm 11}{2}\]
Thus the roots are \[\left( \dfrac{-7+11}{2} \right)\] and \[\left( \dfrac{-7-11}{2} \right)\] \[=\left( \dfrac{4}{2} \right)\] and \[\left( \dfrac{-18}{2} \right)\] = 2 and -9.
Thus x = 2 and x = -9.
So, (x-2) and (x+9).