# Where does the function, f(x)=x^2-6x-7 intersect the function g(x)=-12?

May 7, 2018

They intersect at $x = 1$ and $x = 5$

#### Explanation:

A function is just a way to associate numbers with each others, according to a specified law, or rule. Imagine that you interrogate some robots giving numbers as input, and obtaining numbers as output.

So, two functions intersect if, when "asked the same question", they give the same "answer".

Your first function $f$ takes a number $x$, and gives back that number squared, minus six times that number, minus seven.
The second function $g$, instead, always returns $- 12$, no matter which number $x$ you feed it with.

So, the two functions can only intersect if, for some value $x$, the first function $f$ returns $- 12$.

In formulas, we're looking for a value $x$ such that

$f \left(x\right) = {x}^{2} - 6 x - 7 = - 12 = g \left(x\right)$

If in particular we focus on the middle equality:

${x}^{2} - 6 x - 7 = - 12 \setminus \iff {x}^{2} - 6 x + 5 = 0$

and from here you can use the quadratic formula to solve the equation, obtaining the two solutions ${x}_{1} = 1$, ${x}_{2} = 5$