Question

The difference of the squares of two numbers is 45 the square of the smallest number is 4 times the larger number determine the numbers.

Answer 1

Answer:

Step-by-step explanation:

Given,

• The difference of the squares of two numbers is 45 the square of the smallest number is 4 times the larger number.

To Find,

• The Numbers.

Solution :-

Let the larger number be x.

Then,

Square of the smaller number = 4x

Square of the larger number = x²

Difference of the squares of number = 45

According to the Question,

⇒ x² - 4x = 45

⇒ x² - 4x - 45 = 0

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, - 5.

When x = 9,

Square of smaller number = 4x = 4 × 9 = 36

Smaller number = ± 6

Thus, the numbers are 9 and 6 or 9 and - 6.

When x = - 5

Square of smaller number = 4x = 4 × (- 5) = - 20

Hence, the numbers are 9 and 6 or 9 and - 6.

Answer 2

❍ Let's say that , the Larger number be x & , Square of Smaller number be 4x i.e., The smaller number is 4 times the larger number .

$\qquad \bigstar\:\: \underline{\pmb{\sf \:According \:To \:The \:Question \:\:: }}$

⠀⠀⠀⠀⠀━ The difference of the squares of two numbers is 45 .

$\qquad:\implies \sf \big\{ Larger \: Number \big\}^2 \:- \:\big\{ Squared \:Smaller \:number \:\big\}\:=\:45\:\\\\\\ \qquad:\implies \sf \big\{ x \big\}^2 \:- \:\big\{ 4x \:\big\}\:=\:45\:\\\\\\ \qquad:\implies \sf \:x ^2 \:- 4x \:\:=\:45\:\\\\\\ \qquad:\implies \sf \:x ^2 \:- 4x \:- \:45 \:=\:0\:\\\\\\ \qquad:\implies \sf \:x ^2 \:- 4x \:- \:45 \:=\:0\:\\\\\\ \qquad:\implies \sf \:x ^2 \:- 9x\:+\:5x \:- \:45 \:=\:0\:\\\\\\ \qquad:\implies \sf \:x ( x \:- 9\:)\:+\:5\:(\:x \:- \:9\:) \:=\:0\:\\\\\\\qquad:\implies \sf \: ( x \:- 9\:)\:+\:\:(\:x \:+ \:5\:) \:=\:0\:\\\\\\ \qquad:\implies\underline {\boxed {\pmb{\frak{ \:x \:=\:5 \: or \:9\:}}}}\:\:\bigstar$

Therefore,

⠀⠀⠀⠀When , x = 9 ,

• Smaller Number = $\sf \sqrt{4x} \: =\: \sqrt{36} \: =\:\bf \pm 6 \:$
• Larger Number = 9

⠀⠀⠀⠀When , x = -5 ,

• Smaller Number = $\sf \sqrt{4x} \: =\: \sqrt{-20} \: =\:\bf \pm\sqrt{20} \:$
• Larger Number = -5

⠀⠀⠀⠀⠀∴⠀Hence, The number is 9 and 6 or, 9 and -6 .