**Summary and Review Exercise**

43. We want to fence a field whose length is three times its width, using 100 meters of fencing material. What are the dimensions of the field if we use all the fencing material?

44. Three coffees and two macchiatos cost a total of Birr 17. Two coffees and one macchiato cost a total of Birr 18. What is the individual price for a coffee and a single macchiato?

45. Which of the following is the solution for the system of linear equations?
\[ \left\{
\begin{array}{c}
3x + y = 11 \\
y = x + 2
\end{array}
\right. \]

A) \(x = 3, y = 2\)

B) \(x = 3, y = 5\)

C) \(x = \frac{9}{4}, y = \frac{17}{4}\)

D) \(x = \frac{17}{4}, y = \frac{9}{4}\)

46. The solution for the quadratic equation \(x^2 + 10x = -25\) is:

A) 10

B) -5

C) -10

47. The solution for the absolute value equation \(|-2x + 8| = 12\) is:

A) \(x = 2, x = 10\)

B) \(x = -2, x = 10\)

C) \(x = 2, x = -10\)

D) \(x = -2, x = -10\)

48. Find the values of \(k\) for which the quadratic expression \((x-k)(x-10) = 0\) has integral roots.

49. Find the values of \(k\) such that the equation \(\frac{p}{x+r} + \frac{q}{x-r} = \frac{k}{2x}\) has two equal roots.

50. Find the quadratic equation with rational coefficients when one root is \(\frac{1}{2+\sqrt{5}}\).

51. If the coefficient of \(x\) in the quadratic equation \(x^2 + bx + c = 0\) was taken as 17 instead of 13, its roots were found to be -2 and -15. Find the roots of the original quadratic equation.

52. For what value of \(k\) will both the quadratic equations \(6x^2 - 17x + 12 = 0\) and \(3x^2 - 2x + k = 0\) have a common root?

53. Solve \(\left|x^2 + 2x - 4\right| = 4\).