Advertisements
Advertisements
Question
Find the condition that one of the roots of ax2 + bx + c may be thrice the other
Solution
The given quadratic equation is
ax2 + bx + c = 0 .....(1)
Let α and β be the roots of the equation (1) then
Sum of the roots α + β = 0 .....(2)
Product of the roots αβ = 0 .....(3)
Given that one root is thrice the other
β = 3α
(2) ⇒ α + 3α = `- "b"/"a"`
4α = `- "b"/"a"`
α = `- "b"/(4"a")`
(3) ⇒ α(3α) = `"c"/"a"`
3α2 = `"c"/"a"` ⇒ α2 = `"c"/(3"a")`
`(- "b"/(4"a"))^2 = "c"/(3"a')` ⇒ `"b"^2/(16"a"^2) = "c"/(3"a")`
`"b"^2/(16"a") = "c"/3`
3b2 = 16ac
When is the required condition?
APPEARS IN
RELATED QUESTIONS
Construct a quadratic equation with roots 7 and −3
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`
Find the condition that one of the roots of ax2 + bx + c may be negative of the other
If the equations x2 − ax + b = 0 and x2 − ex + f = 0 have one root in common and if the second equation has equal roots, then prove that ae = 2(b + f)
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
