Ex.4.3 Q7 Quadratic Equations Solutions - NCERT Maths Class 10

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Question

The difference of squares of two numbers is \(180.\) The square of smaller number is \(8\) times the larger number. Find the two numbers.

 Video Solution
Quadratic Equations
Ex 4.3 | Question 7

Text Solution

What is known?

i) Difference of squares of two numbers is \(180.\)

ii) The square of the smaller number is \(8\) times the larger number.

What is Unknown?

Two numbers.

Reasoning:

Let the larger number be \(x.\)

Square of smaller number is \(= 8x.\)

Difference of squares of the two numbers is \(180.\)

Square of larger number - Square of smaller number \(= 180\)

\[{x^2} - 8x - 180 = 0\]

Steps:

\[\begin{align}{x^2} - 8x - 180 &= 0\\{x^2} - 18x + 10x - 180 &= 0\\x(x - 18) + 10(x - 18) &= 0\\
(x - 18)(x + 10) &= 0\\x - 18 &= 0 \quad x + 10 = 0\\x &= 18  \qquad \;\; \;  x = - 10\end{align}\]

If the larger number is \(18,\) then square of smaller number \( = 8 \times 18\)

\[\begin{align}\therefore {\text{ Smaller number }}&{ = \pm \sqrt {8 \times 18} }\\{}&{ = \pm \sqrt {2 \times 2 \times 2 \times 2 \times 3 \times 3} }\\{}&{ = \pm 2 \times 2 \times 3|}\\{}&{ = \pm 12}\end{align}\]

If larger number is \(– 10,\,\) then square of smaller number \( = 8 \times ( - 10) = - 80\)

Square of any number cannot be negative.

\(\therefore x = - 10\) is not applicable.

The numbers are \(18, \,12\) (or) \(18, -12.\)