Advertisements
Advertisements
Question
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 = 2x2 – 15x + 25
⇒ 2x2 – 15x – 3x + 25 + 8 = 0
⇒ 2x2 – 18x + 33 = 0
On company with ax2 + bx + c = 0, we get
a = 2, b = – 18 and c = 33
∴ Discriminant, D = b2 – 4ac
= (–18)2 – 4 × 2(33)
= 324 – 264
= 60 > 0
Therefore, the equation 2x2 – 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`
APPEARS IN
RELATED QUESTIONS
If x=−`1/2`, is a solution of the quadratic equation 3x2+2kx−3=0, find the value of k
Find that non-zero value of k, for which the quadratic equation kx2 + 1 − 2(k − 1)x + x2 = 0 has equal roots. Hence find the roots of the equation.
Find the values of k for which the roots are real and equal in each of the following equation:
k2x2 - 2(2k - 1)x + 4 = 0
Find the value of m for which the equation (m + 4)x2 + (m + 1)x + 1 = 0 has real and equal roots.
`sqrt(3)x^2 + 11x + 6sqrt(3)` = 0
Determine whether the given values of x is the solution of the given quadratic equation below:
6x2 - x - 2 = 0; x = `(2)/(3), -1`.
Find the discriminant of the following equations and hence find the nature of roots: 3x2 + 2x - 1 = 0
(x2 + 1)2 – x2 = 0 has:
If α, β are roots of the equation x2 + px – q = 0 and γ, δ are roots of x2 + px + r = 0, then the value of (α – y)(α – δ) is ______.
If one root of equation (p – 3) x2 + x + p = 0 is 2, the value of p is ______.
