Detailed Solutions to Matched Problems

Solution to Matched Problem 1:

• The perimeter of the rectangle is 60 m, hence
  2x + 2y = 60
  
  
• The area of the rectangle is 200 m², hence
  x*y = 200
  
  
• Solve the equation 2x + 2y = 60 for y.
  y = 30 - x
  
  
• Substitute y in the equation x*y = 200 by the expression for y obtained above.
  x(30 - x) = 200
  
  
• Multiply, group like terms and write the above equation with the right hand side equal to zero.
  -x² +30x - 200 = 0
  
  
• Find the discriminant of the above quadratic equation.
  Discriminant D = b² - 4*a*c = 900 - 800 = 100
  
  
• Use the quadratic formulas to solve the quadratic equation; two solutions
  x1 = [ -b + sqrt(D) ] / 2*a = [ -30 + 10 ] / -2 = 10 m
  
  x2 = [ -b - sqrt(D) ] / 2*a = [ -30 - 10 ] / -2 = 20 m
  
  
• use y = 30 - x found above to find the corresponding value of y.
  y1 = 30 - 10 = 20 m
  y2 = 30 - 20 = 10 m
  
  
• Taking into account the condition x > y, the length x = 20 m and the width y = 10 m.
  
  As an exercise, check the perimeter and the area.

Matched Problem 2: The sum of the squares of two consecutive even real numbers is 52. Find the numbers.

Solution to Problem 2:

• Let x and x+2 be the two consecutive even numbers. The sum of the square of x and x + 2 is equal to 52, hence
  x² + (x + 2)² = 52
  
  
• Expand (x + 2)², group like terms and write the above equation with the right side equal to zero.
  2x² + 4x - 48 = 0
  
  
• Multiply all terms in the above equation by 1/2.
  x² + 2x - 24 = 0
  
  
• Find the discriminant of the above quadratic equation.
  Discriminant D = b² - 4*a*c = 4 + 90 = 100
  
  
• Use the quadratic formulas to solve the quadratic equation; two solutions
  x1 = [ -b + sqrt(D) ] / 2*a = [ -2 + 10 ] / 2 = 4
  x2 = [ -b - sqrt(D) ] / 2*a = [ -2 - 10 ] / 2 = -6
  
  
• First solution to the problem
  first number: x1 = 4
  
  second number: x1 + 2 = 6
  
  
• Second solution to the problem
  first number: x2 = -6
  
  second number: x2 + 2 = -4
  
  As an exercise check that the square of the two numbers, for each solution, is 52.

More references and links on how to Solve Equations, Systems of Equations and Inequalities.