Advertisements
Advertisements
प्रश्न
If one root of k(x − 1)2 = 5x − 7 is double the other root, show that k = 2 or −25
उत्तर
The given quadratic equation is
k(x – 1)2 = 5x – 7
k(x2 – 2x + 1) – 5x + 7 = 0
kx2 – 2kx + k – 5x + 7 = 0
kx2 – (2k + 5)x + k + 7 = 0 ......(1)
Let the roots be α and 2α
Sum of the roots α + 2α = `- "b"/"a"`
3α = `-((-(2"k" +5))/"k")`
3α = `(2"k" + 5)/"k"`
α = `(2"k" + 5)/(3"k")` ......(2)
Product of te roots α(2α) = `"c"/"a"`
2α2 = `("k"+ 7)/"k"`
α2 = `("k" + 7)/(2"k")` .....(3)
Using equation (2) and (3) we have
`((2"k" + 5)/(3"k"))^2 = ("k" + 7)/(2"k")`
`(4"k"^2 + 20"k" + 25)/(9"k"^2) = ("k" + 7)/(2"k")`
`(4"k"^2 + 20"k" + 25)/(9"k") = ("k" + 7)/(2)`
2(4k2 + 20k + 25) = 9k(k + 7)
8k2 + 40k + 50 = 9k2 + 63k
9k2 + 63k – 8k2 – 40k – 50 = 0
k2 + 23k – 50 = 0
k2 + 25k – 2k – 50 = 0
k(k + 25) – 2(k + 25) = 0
(k – 2) (k + 25) = 0
k – 2 = 0 or k + 25 = 0
k = 2 or k = – 25
APPEARS IN
संबंधित प्रश्न
A quadratic polynomial has one of its zeros `1 + sqrt(5)` and it satisfies p(1) = 2. Find the quadratic polynomial
If α and β are the roots of the quadratic equation `x^2 + sqrt(2)x + 3` = 0, form a quadratic polynomial with zeroes `1/α, 1/β`
If the difference of the roots of the equation 2x2 − (a + 1)x + a − 1 = 0 is equal to their product, then prove that a = 2
Find the condition that one of the roots of ax2 + bx + c may be negative of the other
Find the condition that one of the roots of ax2 + bx + c may be thrice the other
Discuss the nature of roots of − x2 + 3x + 1 = 0
Discuss the nature of roots of 4x2 − x − 2 = 0
Discuss the nature of roots of 9x2 + 5x = 0
Without sketching the graph, find whether the graph of the following function will intersect the x-axis and if so in how many points
y = x2 + x + 2
Without sketching the graph, find whether the graph of the following function will intersect the x-axis and if so in how many points
y = x2 − 3x − 7
Write f(x) = x2 + 5x + 4 in completed square form
Solve 2x2 + x – 15 ≤ 0
Solve – x2 + 3x – 2 ≥ 0
Choose the correct alternative:
Find a so that the sum and product of the roots of the equation 2x2 + (a − 3)x + 3a − 5 = 0 are equal is
Choose the correct alternative:
If a and b are the roots of the equation x2 − kx + 16 = 0 and satisfy a2 + b2 = 32, then the value of k is
Choose the correct alternative:
The number of solutions of x2 + |x − 1| = 1 is
Choose the correct alternative:
If 8 and 2 are the roots of x2 + ax + c = 0 and 3, 3 are the roots of x2 + dx + b = 0, then the roots of the equation x2 + ax + b = 0 are
Choose the correct alternative:
If a and b are the real roots of the equation x2 − kx + c = 0, then the distance between the points (a, 0) and (b, 0) is
Choose the correct alternative:
The number of roots of (x + 3)4 + (x + 5)4 = 16 is
