Question
Two number differ by $3$ and their product is $504.$ Find the numbers.
✓
Answer
Let the two numbers be $x$ and $x - 3$ given that $x(x - 3) = 504$
$\Rightarrow x^2 - 3x - 504 = 0$
$\Rightarrow x^2 - 24x + 21x - 504 = 0$
$\Rightarrow x(x - 24) + 21(x - 24) = 0$
$\Rightarrow (x - 24) + 21(x - 24) = 0$
$\Rightarrow (x - 24)(x + 21) = 0$
$\Rightarrow x = 24$ or $x = 21$
Case I: $If x - 3 = 24 - 3 = 21$
Case II: $If x = 21, x = 3 = 24$
$\therefore$ The two numbers are $21, 24$ or $-21, -24$
$\Rightarrow x^2 - 3x - 504 = 0$
$\Rightarrow x^2 - 24x + 21x - 504 = 0$
$\Rightarrow x(x - 24) + 21(x - 24) = 0$
$\Rightarrow (x - 24) + 21(x - 24) = 0$
$\Rightarrow (x - 24)(x + 21) = 0$
$\Rightarrow x = 24$ or $x = 21$
Case I: $If x - 3 = 24 - 3 = 21$
Case II: $If x = 21, x = 3 = 24$
$\therefore$ The two numbers are $21, 24$ or $-21, -24$
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