ICSE Class 10 Maths Chapter 06 Solving Simple Problems Based On Quadratic Equations

Solving (simple) Problems

Based On Quadratic Equations

Introduction

In this chapter, we shall be dealing with word problems based on quadratic equations. (The method of solving a quadratic equation has already been done in previous classes and in the previous chapter).

For solving a word problem based on Quadratic Equation adopt the following steps:

1. Represent the unknown quantity of the problem by variable 'x'.

2. Translate the given statement to form an equation in terms of 'x'.

3. Solve the equation.

Teacher's Note

Word problems teach us to translate real-world situations into mathematical language, a skill we use when budgeting money, planning projects, or calculating distances during travel.

Problems Based on Numbers

Find two natural numbers which differ by 3 and the sum of whose squares is 117.

Solution:

Let the natural numbers be x and x + 3.

\[\therefore x^2 + (x + 3)^2 = 117\]

\[x^2 + x^2 + 6x + 9 = 117\]

\[\Rightarrow 2x^2 + 6x - 108 = 0\]

\[\Rightarrow x^2 + 3x - 54 = 0\]

[Dividing each term by 2]

\[\Rightarrow (x + 9)(x - 6) = 0\]

[Factorising]

\[\Rightarrow x + 9 = 0, \text{ or } x - 6 = 0\]

\[\Rightarrow x = -9, \text{ or } x = 6\]

\[\therefore \text{One number} = 6\]

[Since, -9 is not a natural number]

and other number = 6 + 3 = 9.

\[\therefore \text{Numbers are 6 and 9}\]

Ans.

Teacher's Note

Finding two numbers with specific properties requires setting up equations that represent their relationship, similar to how we might determine dimensions when buying materials for a project.

Five times a certain whole number is equal to three less than twice the square of the number. Find the number.

Solution:

Let the number be x.

Given, five times the number = 3 less than twice the square of the number.

\[\therefore 5x = 2x^2 - 3\]

\[\Rightarrow 2x^2 - 5x - 3 = 0\]

\[\Rightarrow (x - 3)(2x + 1) = 0\]

[Factorising]

\[\Rightarrow x - 3 = 0, \text{ or } 2x + 1 = 0\]

\[\Rightarrow x = 3, \text{ or } x = -\frac{1}{2}\]

\[\therefore \text{Required whole number is 3}\]

Ans.

Sum of two natural numbers is 8 and the difference of their reciprocals is \(\frac{2}{15}\). Find the numbers.

[2015]

Solution:

Let the natural numbers be x and 8 - x

\[\Rightarrow \frac{1}{x} - \frac{1}{8-x} = \frac{2}{15}\]

i.e. \[\frac{8 - x - x}{x(8 - x)} = \frac{2}{15}\]

\[\Rightarrow 2(8x - x^2) = 15(8 - 2x)\]

i.e. \[16x - 2x^2 = 120 - 30x\]

\[\Rightarrow 2x^2 - 46x + 120 = 0\]

i.e. \[x^2 - 23x + 60 = 0\]

\[\Rightarrow x^2 - 20x - 3x + 60 = 0\]

i.e. \[x(x - 20) - 3(x - 20) = 0\]

\[\Rightarrow (x - 20)(x - 3) = 0\]

i.e. \[x - 20 = 0 \text{ or } x - 3 = 0\]

\[\Rightarrow x = 20 \text{ or } x = 3\]

Reject x = 20 as the sum of natural numbers is 8.

\[\therefore x = 3 \text{ and } 8 - x = 8 - 3 = 5\]

\[\Rightarrow \text{Required natural numbers are 3 and 5.}\]

Ans.

Teacher's Note

Problems involving reciprocals often appear in real-world contexts like calculating rates or ratios, such as determining work efficiency or fuel consumption comparisons.

Problems Based on Time and Work

For the same amount of work, A takes 6 hours less than B. If together they complete the work in 13 hours 20 minutes; find how much time will B alone take to complete the work?

Solution:

If B alone takes x hours then A alone takes (x - 6) hours for the same work.

\[\therefore \frac{1}{x-6} + \frac{1}{x} = \frac{3}{40}\]

\[\left[\therefore 13 \text{ hrs. } 20 \text{ min. } = \left(13 + \frac{20}{60}\right) \text{ hrs. } = \frac{40}{3} \text{ hrs.}\right]\]

\[\Rightarrow \frac{x + x - 6}{(x-6)x} = \frac{3}{40}\]

\[\Rightarrow 3x^2 - 18x = 80x - 240\]

i.e. \[3x^2 - 98x + 240 = 0\]

\[\Rightarrow 3x^2 - 90x - 8x + 240 = 0\]

i.e. \[(x - 30)(3x - 8) = 0\]

\[\Rightarrow x = 30, \text{ or } x = \frac{8}{3}\]

i.e. \[x = 30\]

\[\therefore \text{B alone will take 30 hrs. to complete the work.}\]

Ans.

Teacher's Note

Time and work problems model real job situations where we calculate how long it takes individuals or teams to complete tasks, essential for project management and workplace planning.

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