General right triangle

In a right triangle, if a =x+34 and b = x and c= 50, then solve for x. Side c is a hypotenuse. Then discuss the case when a or b is a hypotenuse.

Final Answer:

x =  14
X =  19.7647

Step-by-step explanation:

$$\begin{gathered} a=x+34 \\ b=x \\ c=50 \\ c>a>b \\ {c}^2=a^2+b^2 \\ 50^2=(x+34{)}^2+x^2 \\ -2x^2-68x+1344=0 \\ 2x^2+68x-1344=0 \\ 2...\text{ prime number} \\ 68=2^2\cdot 17 \\ 1344=2^6\cdot 3\cdot 7 \\ \text{GCD}(2,68,1344)=2 \\ {x}^2+34x-672=0 \\ a=1;b=34;c=-672 \\ D=b^2-4ac=34^2-4\cdot 1\cdot (-672)=3844 \\ D>0 \\ {x}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{-34\pm \sqrt{3844}}{2} \\ {x}_{1,2}=\frac{-34\pm 62}{2} \\ {x}_{1,2}=-17\pm 31 \\ {x}_1=14 \\ {x}_2=-48 \\ x>0 \\ x=x_1=14 \end{gathered}$$

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$$\begin{gathered} a>b>c \\ {a}^2=b^2+c^2 \\ (X+34{)}^2=X^2+50^2 \\ 68X-1344=0 \\ 68\cdot X-1344=0 \\ 68X=1344 \\ X=\frac{1344}{68}=19.76470588 \\ X=\frac{336}{17}\doteq 19.764706=19.7647 \end{gathered}$$

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