Advertisements
Advertisements
प्रश्न
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`
Advertisements
उत्तर
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
संबंधित प्रश्न
Determine the nature of the roots of the following quadratic equation:
(x - 2a)(x - 2b) = 4ab
Determine the nature of the roots of the following quadratic equation:
9a2b2x2 - 24abcdx + 16c2d2 = 0
Find the values of k for which the roots are real and equal in each of the following equation:
(3k+1)x2 + 2(k + 1)x + k = 0
Find the values of k for which the roots are real and equal in each of the following equation:
(k + 1)x2 - 2(3k + 1)x + 8k + 1 = 0
If the roots of the equation (a2 + b2)x2 − 2 (ac + bd)x + (c2 + d2) = 0 are equal, prove that `a/b=c/d`.
For what value of k, the roots of the equation x2 + 4x + k = 0 are real?
Find the value of k for which the roots of the equation 3x2 - 10x + k = 0 are reciprocal of each other.
In each of the following determine the; value of k for which the given value is a solution of the equation:
3x2 + 2kx - 3 = 0; x = `-(1)/(2)`
Given that one root of the quadratic equation ax2 + bx + c = 0 is three times the other, show that 3b2 – 16ac.
Find the values of k for which each of the following quadratic equation has equal roots: 9x2 + kx + 1 = 0 Also, find the roots for those values of k in each case.
Find the value(s) of m for which each of the following quadratic equation has real and equal roots: x2 + 2(m – 1) x + (m + 5) = 0
If `(1)/(2)` is a root of the equation `x^2 + kx - (5)/(4) = 0`, then the value of k is ______.
Choose the correct alternative answer for the following sub questions and write the correct alphabet.
What is the value of discriminant for the quadratic equation X2 – 2X – 3 = 0?
State whether the following quadratic equation have two distinct real roots. Justify your answer.
`(x - sqrt(2))^2 - 2(x + 1) = 0`
Let p be a prime number. The quadratic equation having its roots as factors of p is ______.
Solve the equation: 3x2 – 8x – 1 = 0 for x.
The probability of selecting integers a ∈ [–5, 30] such that x2 + 2(a + 4)x – 5a + 64 > 0, for all x ∈ R, is ______.
The sum of all integral values of k(k ≠ 0) for which the equation `2/(x - 1), 1/(x - 2) = 2/k` in x has no real roots, is ______.
If α and β be the roots of the equation x2 – 2x + 2 = 0, then the least value of n for which `(α/β)^n` = 1 is ______.
If 4 is a root of equation x2 + kx – 4 = 0; the value of k is ______.
