EXPLANATION
Considdering the given triangle, we can assevere by the Triangle Proportionality Theorem that there are two similar triangles, and therefore the following relationship is fullfiled:
[tex]\frac{16}{7x}=\frac{4x+4}{8x-3}[/tex]Multiplying both sides by 7x:
[tex]16=7x(4x+4)/(8x-3)[/tex]Multiplying both sudes by 8x-3:
[tex]16(8x-3)=7x(4x+4)[/tex]Applying the distributive property:
128x - 48 = 28x^2 + 28x
Subtracting -(28x^2 +28x) to both sides:
128x - 48 - 28x^2 - 28x = 0
Adding and rearranging terms:
-28x^2 +100x - 48
Solving with the quadratic formula:
[tex]x_1,x_2=\frac{-100\pm\sqrt[]{100^2-4(-28)(-48)}}{2\cdot-28}[/tex]Simplifying:
[tex]x_1,x_2=\frac{-100\pm68}{-56}[/tex]The roots are x_1= 4/7 and x_2 = 3
Just one is the valid option, so considering x=4/7 and replacing in the given equation:
7*4/7 = 4
8* 4/7 - 3 = 11/7
Now, trying with x=3:
7*3 = 21
8*3 -3 = 21 (This is not a viable solution because the Hypotenuse can not be the same as the adjacent leg, so the solution is x=4/7.
The answer is x=4/7