Quadratic Equations - Word Questions

E.g.1

The sum of two numbers is 27 and their product is 50. Find the numbers.
Let one number be x. Then the other number is 50/x.
x + 50/x = 27
X x => x² + 50 = 27x
- 27x => x² - 27x + 50 = 0
(x -25)(x -2) = 0
(x -25) = 0 or (x -2) = 0
x = 25 or x = 2.

E.g.2

The length of a rectangle is 5 cm more than its width and the area is 50cm². Find the length, width and the perimeter.
Let the width be x. Then the length = x + 5.
x(x + 5) = 50
x² + 5x = 50
-50 => x² + 5x - 50 = 0
(x + 10)(x -5) =0
(x + 10) = 0 or (x -5) =0
x = -10 or x = 5 - x = -10 is impossible to be a width
Width = 5cm; so, the length = 10cm.
Perimeter = 30cm.

E.g.3

The three sides of a right-angled triangle are x, x+1 and 5. Find x and the area, if the longest side is 5.
The hypotenuse = 5
x² + (x+1)² = 5² (Pythagoras' Theorem)
x² + x² + 2x + 1 = 25
-25 => x² + x² + 2x - 24 = 0
(x + 6)(x - 4) = 0
(x + 6) = 0 or (x - 4) = 0
x = -6 or x = 4
x = 4;
Area = 0.5 x 3 x 4 = 6cm²

E.g.4

The product of two numbers is 24 and the mean is 5. Find the numbers.
Let one number = x; then the other = 24/x
(x + 24/x)/2 = 5
X 2 => x + 24/x = 10
X x => x² + 24 = 10x
- 10x => x² + -10x + 24 = 0
(x - 6)(x -4) = 0
(x - 6) = 0 or (x -4) = 0
x = 6 or x =4
The numbers are 6 or 4.

E.g.5

The sum of numbers is 9. The squares of the numbers is 41. Find the numbers.
These are quadratic simultaneous equations.
let the numbers be x and y.
x + y = 9
x² + y² = 41
From the first equation, y = (9-x)
Now substitute this in the second equation.
x² + (9-x)² = 41
x² + 81 - 18x + x² = 41
2x² - 16x + 81 = 41
2x² - 16x + 40 = 0
x² - 8x + 20 = 0
(x - 5)(x -4) =0
(x - 5) = 0 or (x -4) =0
x = 5 or x = 4
Substitute in the first equation, y = 5 or 4
The numbers are 5 and 4.

Now, in order to complement what you have just learnt, work out the following questions:

1) The sum of squares of two consecutive even numbers is 244. Find the numbers.
2) The base length of a triangle is 4cm more than its height. The area is 30cm². Find the length of hypotenuse and the perimeter of the triangle.
3) The length of a square is increased by a 5^th so that its new area is 44cm² more than the original value. Find the difference in perimeter of two shapes.
4) The length and width of a rectangular garden are 150m and 120m. A foot path of regular width is added to the boundary of the garden and the total area of the garden becomes 2800m² more than its original area. Find the width of the footpath.
5) A plank of length 10m is cut into two pieces. 5 times the length of the shorter piece is 2 times the length of the longer piece plus 8. Find the length of each piece.
6) The reciprocal of the sum of reciprocals of two numbers is 6. The sum of numbers is 25. Find the numbers.
7) The speed of an ant is (2t + 11), after travelling for t minutes by covering a distance of 12m. Find t.
8) Two chords and a diameter form a triangle inside a circle. The radius is 5cm and one chord is 2cm longer than the other one. Find the perimeter and the area of the triangle.
9) The sum of a number and it reciprocal is 41/20. Find the number.
10) The product of two numbers is 20. The sum of squares is 41. Find the numbers.