# Find whether the following equation have real roots. If real roots exist, find them 12x-3+1x-5=1,x≠32,5 - Mathematics

Sum

Find whether the following equation have real roots. If real roots exist, find them

1/(2x - 3) + 1/(x - 5) = 1, x ≠ 3/2, 5

#### Solution

Given equation is 1/(2x - 3) + 1/(x - 5) = 1, x ≠ 3/2, 5

⇒ (x - 5 + 2x - 3)/((2x - 5),(x - 5)) = 1

⇒ (3x - 8)/(2x^2 - 5x - 10x + 25) = 1

⇒ (3x - 8)/(2x^2 - 15x + 25) = 1

⇒ 3x - 8 = 2x^2 - 15x + 25

⇒ 2x^2 - 15x - 3x + 25 + 8 = 0

⇒ 2x^2 - 18x + 33 = 0

On company with ax^2 + bx + c = 0, we get

a = 2, b = – 18 and c = 33

∴ Discriminant, D = b^2 - 4ac

= (-18)^2 - 4 xx 2(33)

= 324 - 264

= 60 > 0

Therefore, the equation 2x^2 - 18x + 33 = 0 has two distinct real roots

Roots, x = (-b +- sqrt(D))/(2a)

= (-(-18) +- sqrt(60))/(2(2))

= (18 +- 2sqrt(15))/4

= (9 +- sqrt(15))/2

= (9 + sqrt(15))/2, (9 - sqrt(15))/2

Concept: Nature of Roots of a Quadratic Equation
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#### APPEARS IN

NCERT Mathematics Exemplar Class 10