Answer: x = 1/3 or x = 3
(there are two possible values for x)
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The angle markings at the bottom indicate we have congruent angles. In other words, the largest bottom angle (combine the two bottom angles) is bisected. So we'll use the angle bisector theorem.
What that theorem says is that the ratio of the sides is going to be proportional. Specifically the ratio of the adjacent sides to the opposite. This is with the bottom angles as reference angles.
So we have this proportion
(10x-2)/(8x) = (7x)/(5x+3)
Let's solve for x
(10x-2)/(8x) = (7x)/(5x+3)
(10x-2)(5x+3) = 8x*7x
10x(5x+3)-2(5x+3) = 56x^2
50x^2+30x-10x-6 = 56x^2
50x^2+30x-10x-6-56x^2 = 0
-6x^2+20x-6 = 0
6x^2-20x+6 = 0
2(3x^2-10x+3) = 0
3x^2-10x+3 = 0
Now use the quadratic formula to solve for x
a = 3
b = -10
c = 3
x = (-b+-sqrt(b^2-4ac))/(2a)
x = (-(-10)+-sqrt((-10)^2-4*3*3))/(2*3)
x = (10+-sqrt(64))/(6)
x = (10+sqrt(64))/(6) or x = (10-sqrt(64))/(6)
x = (10+8)/(6) or x = (10-8)/(6)
x = 18/6 or x = 2/6
x = 3 or x = 1/3
So that's why the two solutions are
x = 3 or x = 1/3