Question 1174930
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Informally first....<br>
Look at pairs of numbers with a sum of 23 and find a pair with a product of 120.<br>
You can shorten the search by noting that, with a product of 120, one of the numbers must be a multiple of 5.  So the numbers are either 5 and 18 or 15 and 8.<br>
5*18 = 90; 15*8 = 120.  So the two numbers are 15 and 8.<br>
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Now using formal algebra....<br>
let x be one of the two numbers
then 23-x is the other (because their sum is 23)<br>
The product of the two numbers is 120:<br>
{{{x(23-x) = 120}}}
{{{23x-x^2 = 120}}}
{{{x^2-23x+120 = 0}}}<br>
To solve this algebraically, you need to factor the quadratic expression by finding two numbers whose sum is 23 and whose product is 120.<br>
But that's what you had to do with the original problem!<br>
So the formal algebraic approach doesn't get you any closer to the solution of the problem.<br>
However, you should understand the process for solving the problem algebraically.<br>
{{{x^2-23x+120 = 0}}}
{{{(x-15)(x-8) = 0}}}
{{{x = 15}}} or {{{x = 8}}}<br>