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# What numbers can you multiply to get 23716 and add to get 308?

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154

154*154 = 23716

154+154 = 308

You can set up two equations with two unknowns to solve with the quadratic equation. Rearrange one & substitute into the other:

x + y = 308

x*y = 23716

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There are two approaches to this problem, one straightforward and one tricky (but fast). First the straightforward method: call the two numbers x and y, and we have the equations:

(1) xy = 23716

(2) x + y = 308

Using (2) to eliminate y in (1), we arrive at the quadratic equation

x2 - 308x + 23716 = 0

which can be solved using standard techniques. Once x is determined the value of y can be determined from either (1) or (2). So much for the straightforward method.

Now for the tricky-but-fast way. If we inspect the two numbers involved, then we might just be fortunate enough to notice that 23716 is a perfect square:

23716 = 1542.

Furthermore, 154 is half of 308. Therefore the solution is:

154 + 154 = 308,

154 × 154 = 23716.

You can and should check that 154 is indeed a repeated root of the quadratic equation derived above.