Question 707084
Using "x" to represent the original number, "The square of a certain number is 22 less than 13 times the original" translates into:
{{{x^2 = 13x - 22}}}
(Note how in English the "22 less" was first but when we translate it into a mathematical expression it goes at the end. This is usually how it works. If this seems confusing just think of this: If someone asked to figure out 22 less than 45, what would you do? 22-45 or 45-22?<br>
To solve our equation we first want one side to be zero. Subtracting 13x from each side and adding 22 to each side we get:
{{{x^2 - 13x + 22 = 0}}}
Now we factor (or use the Quadratic Formula). This factors fairly easily:
{{{(x-2)(x-11) = 0}}}
Now we use the Zero Product Property which tells us that a product that equals zero, like ours, can only happen if one of the factors is zero. So:
x-2 = 0 or x-11 = 0
Solving these we get:
x = 2 or x = 11<br>
So there are two possible answers to the problem. You can check if you like:
{{{(2)^2 = 13(2) - 22}}}
{{{(11)^2 = 13(11) - 22}}}
Both of these check out.