**Word Problems with Quadratic Equations Examples :**

In this section, we are going to see some examples on solving word problems with quadratic equations.

**Example 1 :**

The diagonal of a rectangular field is 60 meters more than the shorter side. If the longer side is 30 meters more than the shorter side, find the sides of the field.

**Solution :**

Let x be the length of shorter side of the rectangle

Length of diagonal = x + 60

Length of longer side = x + 30

(x + 60)² = x² + (x + 30)²

x² + 60² + 2 x (60) = x² + x² + 2 x (30) + 30²

x² + 3600 + 120 x = 2 x² + 60 x + 900

2 x² - x² + 60 x - 120 x + 900 - 3600 = 0

x² - 60 x - 2700 = 0

x² - 90 x + 30 x - 2700 = 0

x (x - 90) + 30 (x - 90) = 0

(x + 30) (x - 90) = 0

x + 30 = 0 x - 90 = 0

x = -30 x = 90

Breadth of rectangle = 90 m

Length of rectangle = 90 + 30 = 120 m

**Example 2 :**

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

**Solution :**

Let 'x' be the larger number

let 'y' be the smaller number

y² = 8 x

The difference of squares of two numbers = 180

x² - y² = 180

x² - 8 x = 180

x² - 8 x - 180 = 0

x² - 18 x + 10 x - 180 = 0

x (x - 18) + 10 (x - 18) = 0

(x - 18) (x + 10) = 0

x - 18 = 0 x + 10 = 0

x = 18 x = -10

y² = 8 (18)

y = √8 (18)

y = √2 · 2 · 2 · 3 · 3 · 2

y = 2 · 2 · 3

y = 12

Hence the larger number = 18 and smaller number = 12.

**Example 2 :**

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more,it would have taken 1 hour less for the same journey. Find the speed of the train.

**Solution :**

Let x be the speed of the train

If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey

Time = distance/speed

Distance to be covered = 360 km

T₁ = 360/x

T₂ = 360/(x + 5)

T₁ - T₂ = 1 hour

[360/x] - [360/(x + 5)] = 1

360 [ (1/x) - (1/(x + 5))] = 1

360 [ (x + 5 - x)/x(x + 5) ] = 1

360[5/(x² + 5 x)] = 1

1800/(x² + 5 x) = 1

1800 = (x² + 5 x)

x² + 5 x - 1800 = 0

x² - 40 x + 45 x - 1800 = 0

x (x - 40) + 45 (x - 40) = 0

(x - 40) (x + 45) = 0

x - 40 = 0 x + 45 = 0

x = 40 and x = -45

Therefore speed of the train = 40 km/hr

After having gone through the stuff given above, we hope that the students would have understood, word problems with quadratic equations examples.

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