Question

What would be the correct way to solve this?${x}^{2} - 5x - 84$

Answer 1

Answer:

(x + 7 ) ( x - 12 )

Explanation:

We know that if we multiply any two expressions x + a and x + b then we have

$(x+a)(x+b)=x^2+(a+b)x+ab$

Now similarly,

$x^2+(a+b)x+ab=x^2-5x-84$

meaning

$\begin{gathered} a+b=-5 \\ ab=-84 \end{gathered}$

In other words, what are the two numbers that if I add them together I get -5 and If I multiply them I get -84. The answer comes from educated guesses. We guess that if we add 7 and -12 we get 5 and if we multiply then we get -84; therefore,

$\begin{gathered} a=7 \\ b=-12 \end{gathered}$

Hence, the expression can be factored as

$=x^2-5x-84=(x-12)(x+7)$

which is our answer!