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Ber [7]
2 years ago
9

One positive integer is 6 less than twice another. The sum of their squares is 680 . Find the integers.

Mathematics
1 answer:
NemiM [27]2 years ago
4 0

Let's call the two numbers x and y. The first sentence, one is 6 less than twice another, translates to

x = 2y-6

We also know the sum of their squares:

x^2+y^2 = 680

So, we have the following system:

\begin{cases} x = 2y-6 \\ x^2+y^2 = 680 \end{cases}

We can use the expression for x in terms of y from the first equation to turn the second equation into something involving y alone:

x^2+y^2 = 680 \to (2y-6)^2+y^2 = 680

Expand the square:

5y^2-24y+36 = 680

Subtract 680 from both sides:

5y^2-24y-644 = 0

Using the quadratic equation

y_{1,2} = \cfrac{-b\pm\sqrt{b^2-4ac}}{2a}

we find the two solutions

y_1 = -\cfrac{46}{5},\quad y_2 = 14

Since we know that both numbers are integer we can only accept the second solution. It yields the following x value:

x = 2x-6\ \land\ y = 14 \implies x = 2\cdot 14 - 6 = 28-6=22


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