Question

The difference of the squares of two positive numbers is 45. The square of the
smaller number is four times the larger number. Find the numbers​

Answer 1

Answer:

• 9 and 6 are the numbers

Given :

• The difference of the squares of two positive numbers is 45
• The square of the smaller number is four times the larger number

To find :

• Numbers

Solution :

• Let the larger number be x
• Let the smaller number be y

Given that , The differences of the squares of two positive numbers is 45 then ,

• x² + y² = 45 is eq (1)

And also , the square of the smaller number is four times the larger number then

• y² = 4x is eq(2)

According to Question, From eq (1) and eq (2) we get ,

》x² + y² = 45 + y² = 4x

》x² - 4x = 45

》x² - 4x - 45 = 0

》x² - (9 - 5) x - 45 = 0

Now by using factorization method we get,

》x² - 9x + 5x - 45 = 0

》x(x - 9) + 5(x - 9) = 0

》(x - 9) (x + 5) = 0

》x - 9 = 0 or x + 5 = 0

》x = - 9 , x = - 5

》x = 9

Now , putting the value in eq (2) we get,

》y² = 4x

》y² = 4(9)

》y² = 36

》 y = √36

》y = 6

Hence , 9 and 6 are the numbers

Answer 2

Given:-

• Difference of the squares of two positive numbers is 45.

• Square of the smaller number is four times the larger number.

To Find:-

• The Numbers

Criteria used:-

• Middle Term Split

Solution:-

Let the larger and smaller number be x and y respectively.

According to question,

➠ x² - y² = 45 ____ {1}

And also,

➠ y² = 4x ____ {2}

Putting this value in {1},

➠ x² - 4x = 45

We can also write it like this,

➠ x² - 4x - 45 = 0

By Middle term split,

➠ x² - (9 - 5)x - 45 = 0

➠ x² - 9x + 5x - 45 = 0

➠ x(x - 9) + 5(x - 9) = 0

➠ (x - 9) (x + 5) = 0

➠ x - 9 = 0 and x + 5 = 0

➠ x = 9 and x = -5

As, it's given that the two numbers are positive So, we ignore the negative term. Hence, x = +9.

Putting the value of x in {2},

➠ y² = 4x

➠ y² = 4 × 9

➠ y² = 36

➠ y = √36

➠ y = ± 6

Here also we will ignore negative term. Hence, y = +6.

Hence, the two numbers are 9 and 6.

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