Question

Clara writes the equation (x – 13)(x + 8) = 196 to solve for the missing side length of a triangle represented by the factor x + 8. What is the missing side length represented by x + 8 units of the triangle?

Answer 1

The answer is 28.

(x – 13)(x + 8) = 196
x * x + x * 8 - 13 * x - 13 * 8 = 196
x² + 8x - 13x - 104 = 196
x² - 5x - 104 - 196 = 0
x² - 5x - 300 = 0

The general quadratic formula is ax² + bx + c = 0
In our equation: a = 1, b = -5, c = -300

$x_{1,2} = \frac{-b+/- \sqrt{ b^{2}-4ac } }{2a} = \frac{-(-5)+/- \sqrt{ (-5)^{2}-4*1*(-300) } }{2*1}=\frac{5+/- \sqrt{ 25+1200 } }{2}= \\ \\ = \frac{5+/- \sqrt{1225} }{2}= \frac{5+/-35}{2} \\ \\ x_1= \frac{5+35}{2}= \frac{40}{2} =20 \\ \\ x_2= \frac{5-35}{2}= \frac{-30}{2} =-15$

So, missing length is either:
x1 + 8 = 20 + 8 = 28
or
x2 + 8 = -15 + 8 = -7

Since, length of a triangle cannot be negative, the missing side length is 28