Question

The difference between two numbers is 3. The square of the smaller number is equal to 4 times the larger number.
Find the two numbers.

Answer 1

Answer:

Numbers are x and x + 3; x2 = 4(x + 3) ie 4x + 12,

so there is a quadratic equation x2 - 4x - 12 = 0

This factorises as (x - 6)(x + 2) giving solutions of x = 6 (6 x 6 = 4 x (6 + 3))

or -2 (-2 x -2 = 4 x (-2 + 3))

To summarise: The alternative answers are 6 and 9 or -2 and 1

Answer 2

Answer:

The two numbers are 1 and - 2 or 9 and 6 respectively.

Step-by-step explanation:

Given:

• The difference between two numbers is 3.
• The square of the smaller number is equal to 4 times the larger number.

To find:

• The two numbers.

Solution:

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

The difference between two numbers is 3.

• x - y = 3
• y = x - 3. ...1

Now,

The square of the smaller number is equal to 4 times the larger number which means 4 multiplied by larger number is equal to square of smaller number.

• y² = 4 × x

Substituting the value of eq1 in the above equation, we get:

• (x - 3)² = 4x

Expanding the given term,

• x² + 3² - 2 × x × 3 = 4x
• x² + 9 - 6x = 4x
• x² + 9 - 6x - 4x = 0
• x² - 10x + 9 = 0
• x² - 9x - 1x + 9 = 0
• x(x - 9) - 1(x - 9) = 0
• (x - 1) (x - 9) = 0

Take,

• x - 1 = 0
• x = 1

Now, take

• x - 9 = 0
• x = 9

Now, find the value of y in both the cases,

• y = 1 - 3
• y = - 2

• y = 9 - 3
• y = 6

∴ The required numbers are 1 and - 2 or 9 and 6 respectively.

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