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

## 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.

## 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:**

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

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

Therefore, the smaller number

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

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) \(18, -12.\)