Question

28. 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

$\large\underline{\sf{Solution-}}$

↝ Let assume that

• Larger number be x

• Smaller number be y.

According to statement,

↝ The difference of squares of two numbers is 180.

$\purple{\rm :\longmapsto\: {x}^{2} - {y}^{2} = 180 - - - (1)}$

According to second condition,

↝ The square of smaller number is 8 times the larger number.

$\red{\rm :\longmapsto\: {y}^{2} = 8x - - - (2)}$

On substituting the value of equation (2), in (1) we get

$\rm :\longmapsto\: {x}^{2} - 8x = 180$

$\rm :\longmapsto\: {x}^{2} - 8x - 180 = 0$

$\rm :\longmapsto\: {x}^{2} - 18x + 10x- 180 = 0$

$\rm :\longmapsto\: x({x} - 18) + 10(x- 18) = 0$

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

$\rm :\implies\:x = 18 \: \: \: or \: x = - 10$

Two cases arises

$\red{\bf :\longmapsto\:When \: x = \: 18}$

Put value of x = 18 on equation (2) we get

$\rm :\longmapsto\: {y}^{2} = 8 \times 18$

$\rm :\longmapsto\: {y}^{2} = 144$

$\rm :\longmapsto\:y = \: \pm \: \sqrt{144}$

$\bf\implies \:y \: = \: \pm \: 12$

$\red{\bf :\longmapsto\: \: When \: x = \: - 10}$

On substituting the value of x = - 10 in equation (2), we get

$\rm :\longmapsto\: {y}^{2} = 8 \times ( - 10)$

$\rm :\longmapsto\: {y}^{2} = - 80$

which is not possible.

Hence,

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:Numbers \: are-\begin{cases} &\bf{18, \: 12} \\&\bf{ \: \: \: \: \: or} \\ &\bf{18, \: - \: 12} \end{cases}\end{gathered}\end{gathered}$

Additional Information :

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

• If Discriminant, D > 0, then roots of the equation are real and unequal.

• If Discriminant, D = 0, then roots of the equation are real and equal.

• If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

• Discriminant, D = b² - 4ac