## SOLVING WORD PROBLEMS ON QUADRATIC EQUATIONS

Solving Word Problems on Quadratic Equations :

In this section, we will solve some examples expressed in words and some problems describing day to day life situation involving quadratic equation.

To solve a given word problem involving quadratic equations, we follow the steps given below.

Step 1 :

Convert the word problem to a quadratic equation form.

Step 2 :

Solve the quadratic equation obtained in any one of the methods

Step 3 :

Relate the mathematical solution obtained to the statement asked in the question.

## Solving Word Problems on Quadratic Equations - Examples

Example 1 :

Find two consecutive positive even integers whose squares have the sum 340.

Solution :

Let x and (x + 2) be two positive even integers.

Sum of their squares  =  340

x2 + (x + 2)2  =  340  ---(1)

Expanding (x + 2)2 using the the algebraic identity

(a + b)2  =  a2 + 2ab + b2

(x + 2)2  =  x2 + 2 (x) (2) + 22

(x + 2)2  =  x2 + 4x + 4

By applying the expansion in (1), we get

x2 + x² + 4x + 4  =  340

2x² + 4x - 336  =  0

By dividing the entire equation y 2, we get

x2 + 2x - 168  =  0

By factoring, we get

(x - 12) (x + 14)  =  0

 x - 12  =  0x  =  12 x + 14  =  0x  =  -14

Since the required number is even positive integers, we choose positive integer.

If x  =  12, then x + 2  ==>  14

Therefore two positive even integers are 12 and 14.

Verification :

Sum of their squares is 340

=  122 + 142

=  144 + 196

=  340

Example 2 :

The sum of two squares of three consecutive natural numbers is 194. Determine the numbers.

Solution :

Let x , x + 1 and x + 2 be three consecutive natural numbers

Sum of their squares  =  194

x2 + (x + 1)2 + (x + 2)2  =  194

(x + 1)2  =  x2 + 2x + 1

(x + 2)2  =  x2 + 4x + 4

x2 + x2 + 2x + 1 + x2 + 4x + 4  =  194

3x2 + 6x + 5  =  194

3x2 + 6x + 5 - 194  =  0

3x2 + 6x - 189  =  0

Dividing the entire equation by 3, we get

x2 + 2x - 63  =  0

(x + 9) (x - 7) = 0

 x + 9  =  0x  =  -9 x - 7  =  0x  =  7

If x  =  7

then x + 1 ==> 7 + 1 ==> 8

and x + 2 ==> 7 + 2 ==> 9

Therefore the required numbers are 7, 8, 9

Verification :

The sum of two squares of three consecutive natural numbers is 194

=  72 + 82 + 92

=  49 + 64 + 81

=  194

After having gone through the stuff given above, we hope that the students would have understood how to solve word problems using quadratic equations.

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