Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
2x2 − (a + 1)x + (a − 1) = 0
Let α + β be the roots
⇒ α + β = `(α + 1)/2` and αβ = `(α - 1)/2`
⇒ (α + β)2 = (αβ)2
(i.e.,) (α + β)2 – 4αβ = (αβ)2
⇒ `((α + 1)/2)^2 4((α - 1)/2) = ((α - 1)/2)^2`
`(α^2 + 2α + 1)/4 - 2(α - 1) = (α^2 - 2α+ 1)/4`
α2 + 2α + 1 − 8(α − 1) = α2 − 2α + 1
α2 + 2α + 1 − 8α + 8 − α2 + 2α − 1 = 0
− 4a + 8 = 0 ⇒ 4a = 8
α = `8/4` = 2
APPEARS IN
संबंधित प्रश्न
Construct a quadratic equation with roots 7 and −3
A quadratic polynomial has one of its zeros `1 + sqrt(5)` and it satisfies p(1) = 2. Find the quadratic polynomial
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
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 − x2 + 3x + 1 = 0
Discuss the nature of roots of 4x2 − x − 2 = 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
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:
The equation whose roots are numerically equal but opposite in sign to the roots of 3x2 − 5x − 7 = 0 is
Choose the correct alternative:
The number of roots of (x + 3)4 + (x + 5)4 = 16 is
