length of the shorter leg = k
length of the longer leg = 2k - 3
length of the hypothenus = 5 + 2k
to solve this problem, we have to use pythagorean theorem
pythagorean theorem states that
[tex]\text{hyp}^2=\text{adj}^2+\text{opp}^2[/tex]hyp = hypothenus
adj = adjacent
opp = opposite
now let's plug in our variables into the equation
[tex](5k+2)^2=(2k-3)^2+k^2[/tex]this is an equation to find the value of k
we can further simplify this to get a quadratic equation
[tex]\begin{gathered} (5k+2)^2=(2k-3)^2+k^2_{} \\ 10k^2+20k+4=(4k^2-12k+9)+k^2 \\ 10k^2+20k+4=4k^2-12k+9+k^2 \\ \text{collect like terms} \\ 10k^2+20k+4-4k^2+12k-9-k^2 \\ (10-4-1)k^2+(20+12)k-9+4_{} \\ 5k^2+32k-5=0 \end{gathered}[/tex]the above written equation can be used to solve for k
note: in opening the bracket
[tex]\begin{gathered} (5k+2)^2=a^2+2ab+b^2 \\ \text{that is the how the bracket opens} \\ a=5 \\ b=2 \end{gathered}[/tex]
