Question

the difference of squares of two numbers is 180 the square of the smaller number is 8 times the larger number find the two numbers

Answer 1

Answer:

Let the larger number = x

Then the square of the smaller number = 8 times the larger number = 8x

and the square of the larger numbe r = x

According to the question,

x - 8x = 180

=> x - 8x - 180 = 0

=> x - 18x + 10x - 180 = 0

=> x(x - 18) + 10(x - 18) = 0

=> (x - 18) (x + 10) = 0

=> x - 18 = 0 or x + 10 = 0

=> x = 18 or x = -10

Thus, the larger number = 18 or -10

Then, the square of the smaller number = 8(18) or 8(-10)

= 144 or -80

The square of a number can't be negative, so, the square of smaller number = 144

Hence, the smaller number = sqrt(144) = 12

The numbers are 12 and 18

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Step-by-step explanation:

Answer 2

Appropriate Question:

The difference of squares of two two numbers is 180. If the square of the smaller number is 8 tumes the larger number, find the two numbers.

Given:

Difference of squares of two numbers is 180.

the smaller number is 8 times the larger number.

To Find:

Find The Two Numbers.

Answer:

Let The Smaller Number be x and, the larger number = y.

According To The Statment,

$\tt{ {y}^{2} - {x}^{2} = 180 \: \: [y > x]} \: \: \: \: \sf{...(i)}$

$: \implies \tt{Also, \: {x}^{2} = 8y } \: \: \: \sf{ ...(ii)}$

Substituting The Value Of (ii) in (i), We Get, $\tt{ {y}^{2} - 8y = 180 }$

$: \implies \tt{ {y}^{2} - 8y - 180 = 0 } \\ : \implies \tt{ {y}^{2} - 18y + 10y - 180 = 0} \\ : \implies \tt{y(y - 18) + 10(y - 18) = 0} \\ : \implies \tt{(y + 10)(y - 18) = 0}: \implies \tt{y + 10 = 0} \: \: \: \: \: \bf{or} \: \: \: \tt{ y - 18 = 0} \\ : \implies \tt{y = - 10} \: \: \: \bf{or} \: \: \: \tt{y = 18}$

Now,

$\tt{if \: y = - 10 \: is \: rejected.} \\ \tt{then \: {x}^{2} = 8 \times ( - 10) } \: \: \: \: [Using(ii)] \\ : \implies \tt{ {x}^{2} = - 80 } \\ : \implies \tt{x= \pm \: \sqrt{ - 80} } \: \: \: \: [Not \: Real \: Root]\\ : \implies \tt{So, \: y = - 10 \: is \: rejected.} \\ : \implies \tt{If \: y=18, Then, \: {x}^{2} = 8 \times 18 \: \: \: \: [Using \: (ii)] } \\ : \implies \tt{ {x}^{2} = 144} \\ : \implies \tt{x = \pm12}$

Hence, Two Numbers are (12, 18) or (-12, 18) .