The perimeter of a triangle is $15 x^{2} - 23 x + 9$ and two of its sides are $5 x^{2} + 8 x - 1$ and $6 x^{2} - 9 x + 4.$ Find the third side.

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Solution

The perimeter of triangle $= 15 x^{2} - 23 x + 9$
Length of one side $= 5 x^{2} + 8 x - 1$
Length of other side $= 6 x^{2} - 9 x + 4$
Sum of the lengths of the two sides $= (5 x^{2} + 8 x - 1) + (6 x^{2} - 9 x + 4) = 11 x^{2} - x + 3$
∴ Length of the third side = Perimeter of the triangle − Sum of the other two sides
$= (15 x^{2} - 23 x + 9) - (11 x^{2} - x + 3)$
$= 15 x^{2} - 23 x + 9 - 11 x^{2} + x - 3$
$= 4 x^{2} - 22 x + 6$

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The perimeter of a triangle is 15x2−23x+9 and two of its sides are 5x2+8x−1 and 6x2−9x+4. Find the third side.

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