Question

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

Answer 1

$\large\underline\mathrm\red{Given:-}$

• $\large\mathrm{The \: difference \: of \: square \: of \: the \: number \: is \: 180.}$

• $\large\mathrm{Square \: of \: smaller \: number \: is \: 8 \: times \: the \: larger \: number}$

$\large\underline\mathrm\red{To \: find}$

• $\large\mathrm{The \: larger \: number}$

$\large\underline\mathrm\red{Solution}$

• $\large\mathrm{Let \: the \: larger \: number \: be \: x}$

• $\large\mathrm{Let \: the \: smaller \: number \: be \: y}$

$\large\underline\mathrm\red{A/Q}$

$\large\mathrm{The \: square \: of \: the \: smaller \: number \: is \: 8 \: time \: the \: larger \: number}$

$\large\mathrm{⟹ {y}^{2} = 8x}$_______(1)

$\large\mathrm{The \: difference \: of \: square \: of \: two \: number \: is \: 180°}$

$\large\mathrm{⟹ {x}^{2} - {y}^{2} = {180}}$______(2)

$\large\mathrm{Substitution \: value \: of \: {y}^{2} = 8x \: from \: (1) \: and \: (2).}$

$\large\mathrm{⟹ {x}^{2} - 8x = 180.}$

$\large\mathrm{So, \: degree \: of \: the \: Eq. \: is 2 \: so \ it's \: a \: Quadratic \: Equation.}$

$\large\underline\mathrm\red{Now}$,

$\large\mathrm{Solving \: the \: quadratic \: equation \: by \: factorisation \: method.}$

$\large\mathrm{⟹ {x}^{2} - 8 - 180 = 0}$

$\large\mathrm{⟹ {x}^{2} + 18x - 10x - 180 }$

$\large\mathrm{⟹ x(x - 18) + 10(x - 18)}$

$\large\mathrm{⟹ (x + 10) \: (x - 18)}$

$\large\mathrm{⟹ x = - 10 \: Or \: x = 18}$

$\large\mathrm{When \: x = - 10 \: then \: value \: of \: is:-}$

$\large\mathrm{⟹ {y}^{2} = 8x}$

$\large\mathrm{⟹ {y}^{2} = 8 × (-10)}$

$\large\mathrm{⟹ {y}^{2} = - 18}$

$\large\mathrm{⟹ y = +,- \: (root \: -80)}$

$\large\mathrm{So \: we \: can't \: the \: negative \: number \: in \: root \: so \: the \: value \: of \: y = \: root \: -80. \: is \: not \: possible.}$

$\large\mathrm{Now, \:when \: x = 18 \: then \: value \: of \: y \: :-}$

$\large\mathrm{⟹ {y}^{2} = 8x}$

$\large\mathrm{⟹ {y}^{2} = 8 × 18}$

$\large\mathrm{⟹ {y}^{2} = 144}$

$\large\mathrm{⟹ y = \: (root \: 144)}$

$\large\mathrm{⟹ y = +,- 12}$

$\large\underline\mathrm\red{Now}$,

$\large\mathrm\red{The \: two \: number \: are:-}$

$\large\mathrm{≫ Smaller \: number \: = \: 12 \: Or \: - 12}$

$\large\mathrm{≫ Large \: number \: = \: 18}$

__________________________________

Answer 2

Answer:

this is your ans

I hope it helps