Hi, there-
Here is how to solve this Pythagorean Equation for x, and the side lengths of the right triangle.
x^2 + (2x+6)^2 = (2x+9)^2
This equation represents a right triangle with legs of length, x and 2x+6, and hypotenuse of length
2x+9.
Square the two binomials to clear the parentheses.
x^2 + 4x^2 +24x + 36 = 4x^2 +36x + 81
Subtract 4x^2 from both sides of the equation.
x^2 +24x + 36 = 36x + 81
Subtract 36x from both sides of the equation.
x^2 -12x + 36 = 81
Subtract 91 from both sides of the equation.
x^2 -12x -45 = 0
Factor this quadratic polynomial. We want two numbers whose product is -45 and whose sum is -12.
The numbers are -15 and 3.
(x - 15)(x + 3) = 0
x - 15 = 0 OR x + 3 = 0
x = 15 OR x = -3
Both these values for x make the original equation true, but x = -3 does not make sense if we are
talking about side lengths of triangles (you can't have a triangle with a side length of -3.)
Therefore, x = 15.
2x + 6 = 2(15) +6 = 36.
2x + 9 = 2(15) + 9 = 39.
The legs of the right triangle have lengths 15 and 36. (This is where the 36 came from!) The
hypotenuse has a length of 39.
Hope this helps. Feel free to email me if you have questions about this,
Ms.Figgy
math.in.the.vortex@gmail.com
