Detailed Solutions to Matched Problems
Detailed solutions to the matched problems in Quadratic Equations  Problems (1) are presented.
 Matched Problem 1: A rectangle has a perimeter of 60 m and an area of 200 m^{2}. Find the length x and width y, x > y, of the rectangle.
Solution to Matched Problem 1:

The perimeter of the rectangle is 60 m, hence
2x + 2y = 60

The area of the rectangle is 200 m^{2}, 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.
x^{2} +30x  200 = 0

Find the discriminant of the above quadratic equation.
Discriminant D = b^{2}  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
x^{2} + (x + 2)^{2} = 52

Expand (x + 2)^{2}, group like terms and write the above equation with the right side equal to zero.
2x^{2} + 4x  48 = 0

Multiply all terms in the above equation by 1/2.
x^{2} + 2x  24 = 0

Find the discriminant of the above quadratic equation.
Discriminant D = b^{2}  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.

