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
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) $$18, -12.$$

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