Learn Exercise 4.2 with Free Lessons & Tips

Let X and Y be the two numbers.Given,

X+Y=27 ---- (1)

XY=182-----(2)

From(1)

Y=27-X

Substituting in (2)

(27-X)X = 182

27X-X^2 =182

X^2-27X+182=0

On solving the quadratic equation

(X-13)(X-14)=0

X=13 or X=14

Therefore,

X=13 implies Y=14

Or

X=14 implies Y=13

Let the first integer be x & the second be

(x + 1).

ATQ, x² + (x + 1)² = 365

or x² + x² + 2x +1 = 365 

or 2x² + 2x - 364 = 0 or x² + x - 182 = 0

or x² + 14x - 13x - 182 = 0

or x (x + 14) -13 (x + 14) = 0

or (x - 13) (x + 14) = 0

x = 13 or x = -14

Since the value of x can't be negative, therefore, the integers are 13 & 14.

(i) x² - 3x - 10 = 0 or x² - 5x + 2x - 10 = 0

or x (x - 5) + 2 (x - 5) = 0 

or (x + 2) (x - 5) = 0 x = - 2 & x= + 5

(ii) √2x² + 7x + 5√2 = 0

or √2x² + 2x + 5x + 5√2 = 0

or √2x(x + √2) + 5(x + √2) = 0

or (√2x + 5) (x + √2) = 0

x = -5/√2 & -√2

(iii) 2x² + x - 6 = 0 or 2x² + 4x - 3x - 6 = 0 

or 2x(x + 2) - 3(x + 2) = 0

or (2x - 3) (x + 2) = 0

x = 3/2 & -2

(iv) 2x² - x + 1/8 = 0 or 16x² - 8x + 1 = 0

or (4x - 1) (4x - 1) = 0

x = 1/4 & 1/4

(v) 100x² - 20x + 1 = 0

or (10x - 1) (10x - 1) = 0

x = 1/10 & 1/10

Let the Base be = x cm

then Altitude = (x-7) cm

Since it is a right triangle, we can apply the Pythagorean theorem.

x(x-12)+5(x-12)=0

(x-12)(x+5)=0

x=12  or  x=-5

Since x is base which cannot be negative

Hence x (base) = 12 cm

and  Altitude = 12-7 = 5 cm