[php] Find what 2 numbers add to something and multiply to something

Hey so I'm making a factoring program and I'm wondering if anyone can give me any ideas on an efficient way to find what two numbers multiple to a specified number, and also add to a specified number.

for example I may have

(a)(b) = 6

a + b = 5

So essentially i just need a way to find the a and b values. In this case they would be 2 and 3.

Can anyone give me any ideas on where to start? Negative numbers must also be considered for use.

That's basically a set of 2 simultaneous equations:

x*y = a
X+y = b

(using the mathematical convention of x and y for the variables to solve and a and b for arbitrary constants).

But the solution involves a quadratic equation (because of the x*y), so depending on the actual values of a and b, there may not be a solution, or there may be multiple solutions.

With the multiplication, I recommend using the modulo operator (%) to determine which numbers divide evenly into the target number like:

$factors = array();
for($i = 0; $i < $target; $i++){
    if($target % $i == 0){
        $temp = array()
        $a = $i;
        $b = $target / $i;
        $temp["a"] = $a;
        $temp["b"] = $b;
        $temp["index"] = $i;
        array_push($factors, $temp);
    }
}

This would leave you with an array of factors of the target number.

Here is how I would do that:

$sum = 5;
$product = 6;

$found = FALSE;
for ($a = 1; $a < $sum; $a++) {
  $b = $sum - $a;
  if ($a * $b == $product) {
    $found = TRUE;
    break;
  }
}

if ($found) {
  echo "The answer is a = $a, b = $b.";
} else {
  echo "There is no answer where a and b are both integers.";
}

Basically, start at $a = 1 and $b = $sum - $a, step through it one at a time since we know then that $a + $b == $sum is always true, and multiply $a and $b to see if they equal $product. If they do, that's the answer.

See it working

Whether that is the most efficient method is very much debatable.