Question

the difference of squares of two number is 180.the square of the smaller number is 8 times the larger number. find the two number
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Answer 1

Answer:

Let ‘X’ and ‘Y’ be the two numbers.

Then By Question, We have,

X² - Y² = 180 ——————(1)

Y² = 8X ———————— (2)

Now, substituting the value of Y² from equation (2) in equation (1), we get,

X² - Y² = 180

X² - 8X = 180

X² - 18X + 10X = 180

X(X-18)+10(X-18) = 0

(X-18) (X+10) = 0

Now, Either,

X-18 = 0 or X+10 = 0

X = 18 or X = -10 (ignored because negative)

Therefore, X = 18

Again, we know that,

Y² = 8X

Y = √8X

Y = √8*18

Y = √144

Y = 12

Hence, the two numbers are 12 and 18.

Answer 2

Solution:

Let the two numbers be x and y ; the smaller number be y and the larger number be x By given data,

$x {}^{2} - y {}^{2} = 180 \: \: \: \: .............(1)$

And,

$y {}^{2} = 8x$

$(1) = > x {}^{2} - 8x = 180 \\ \: \: \: \: \: \: \: \: \: \: \: \: \: = > x ^{2} { - 8x - 180} = 0$

By factorization we get, -180< -18 and 10

$(x - 18)(x + 10) = 0 \\ x = 18 \: and \: x = - 10$

$when \: x = 18 \\ (1) = > 18 { }^{2} - y {}^{2} = 180 \\ \: \: \: \: \: \: \: = > 324 - y {}^{2} = 180 \\ \: \: \: \: \: \: \: \: \: \: \: = > - y {}^{2} = 180 - 324 \\ \: \: = > - y {}^{2} = - 144 \\ = > y = \sqrt{144} \\ = > y = 12$

$when \: x = - 10 \\ (1) = > ( - 10) { }^{2} - y {}^{2} = 180 \\ = > 100 - y {}^{2} = 180 \\ = > - y {}^{2} = 180 - 100 \\ = > - y {}^{2} = 180 \\ = > y {}^{2} = - 180 \\ = > y = \sqrt{ - 180 \: } \: which \: is \: imaginary \: .hence \: y \: is \: not \: equal \: to \: - 10$

Therefore, the two numbers are x= 18 and y= 12