Answer
$a = 3.83$
Work Step by Step
We can find rearrange the equation as a quadratic equation:
$c^2 = a^2+b^2$
$(a+4)^2 = a^2+(a+3)^2$
$a^2+8a+16 = a^2+(a^2+6a+9)$
$a^2+8a+16 = 2a^2+6a+9$
$a^2-2a-7=0$
We can use the quadratic formula to find the solutions of the equation:
$a = \frac{-B \pm \sqrt{B^2-4AC}}{2A}$
$a = \frac{-(-2) \pm \sqrt{(-2)^2-(4)(1)(-7)}}{(2)(1)}$
$a = \frac{2 \pm \sqrt{4+28}}{2}$
$a = \frac{2 \pm \sqrt{32}}{2}$
$a = \frac{2 - \sqrt{32}}{2}~~$ or $~~a = \frac{2 + \sqrt{32}}{2}$
$a = -1.83~~$ or $~~a = 3.83$
Since the length $a$ must be positive, the value of $a$ is $~~3.83$
