Q. 6

# Show that the roots of the equation

(x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0 are always real and they cannot be unless a = b = c.

Answer :

(x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0

⇒ x(x – b) – a(x – b) + x(x – c) – b(x – c) + x(x – a) – c(x – a) = 0

⇒ x^{2} – bx – ax + ab + x^{2} – cx – bx + bc + x^{2} – ax – cx + ac = 0

⇒ 3x^{2} – 2x(a + b + c) + ab + bc + ac = 0

D = b^{2} – 4ac

D = (a + b + c)^{2} – 4(3)(ab + bc + ac) = 0

D = 4(a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca – 3ab – 3bc – 3ca)

D = 4(a^{2} + b^{2} + c^{2} – ab – bc – ca)

D = 2[(a –b)^{2} + (b – c)^{2} + (c – a)^{2}]

Which is always greater than zero so the roots are real.

Roots are equal if D = 0

i.e. (a – b)^{2} + (b + c)^{2} + (c – a)^{2} = 0

since sum of three perfect square is equal to zero so each of them separately equal to zero.

So, a – b = 0, b – c = 0, c – a = 0

a = b , b = c, c = a

so, a = b = c.

Rate this question :

Determine the nature of the roots of the equation.

x^{2} – 8x + 12 = 0

Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.

3, 1

Tamil Nadu Board - MathFind a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.

Tamil Nadu Board - Math

Solve the following quadratic equations by completing the square.

=3x + 2

Tamil Nadu Board - MathFind a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.

0, 4

Tamil Nadu Board - MathFind the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.

4x^{2} + 8x

Let b = a + c. Then the equation ax^{2} + bx + c = 0 has equal roots, if

A chess board contains 64 equal squares and the area of each square is 6.25 cm^{2}. An order around the board is 2 cm wide. Find the length of the side of the chess board