Advertisements
Advertisements
प्रश्न
Show that the equation 2(a2 + b2)x2 + 2(a + b)x + 1 = 0 has no real roots, when a ≠ b.
Advertisements
उत्तर
The quadric equation is 2(a2 + b2)x2 + 2(a + b)x + 1 = 0
Here,
a = 2(a2 + b2), b = 2(a + b) and c = 1
As we know that D = b2 - 4ac
Putting the value of a = 2(a2 + b2), b = 2(a + b) and c = 1
D = {2(a + b)}2 - 4 x (2(a2 + b2)) x (1)
= 4(a2 + 2ab + b2) - 8(a2 + b2)
= 4a2 + 8ab + 4b2 - 8a2 - 8b2
= 8ab - 4a2 - 4b2
= -4(a2 - 2ab + b2)
= -4(a - b)2
We have,
a ≠ b
a - b ≠ 0
Thus, the value of D < 0
Therefore, the roots of the given equation are not real
Hence, proved
APPEARS IN
संबंधित प्रश्न
If -5 is a root of the quadratic equation 2x2 + px – 15 = 0 and the quadratic equation p(x2 + x)k = 0 has equal roots, find the value of k.
Find the nature of the roots of the following quadratic equation. If the real roots exist, find them:
`3x^2 - 4sqrt3x + 4 = 0`
Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2? If so, find its length and breadth.
Determine the nature of the roots of the following quadratic equation:
2(a2 + b2)x2 + 2(a + b)x + 1 = 0
Find the value of the discriminant in the following quadratic equation :
`4 sqrt 3 "x"^2 + 5"x" - 2 sqrt 3 = 0`
Determine the nature of the roots of the following quadratic equation :
(x - 1)(2x - 7) = 0
Solve the following quadratic equation using formula method only :
x2 +10x- 8= 0
Solve the following quadratic equation using formula method only
3a2x2 +8abx + 4b2 = 0, a ≠ 0
Find if x = – 1 is a root of the equation 2x² – 3x + 1 = 0.
`10x -(1)/x` = 3
Find the value of k for which the given equation has real roots:
kx2 - 6x - 2 = 0
Solve for x : `9^(x + 2) -6.3^(x + 1) + 1 = 0`.
Choose the correct answer from the given four options :
If the equation 3x² – kx + 2k =0 roots, then the the value(s) of k is (are)
If roots of a quadratic equation 3y2 + ky + 12 = 0 are real and equal, then find the value of ‘k’.
α and β are the roots of 4x2 + 3x + 7 = 0, then the value of `1/α + 1/β` is:
Mohan and Sohan solve an equation. In solving Mohan commits a mistake in constant term and finds the roots 8 and 2. Sohan commits a mistake in the coefficient of x. The correct roots are:
If the coefficient of x2 and the constant term have the same sign and if the coefficient of x term is zero, then the quadratic equation has no real roots.
Find the nature of the roots of the quadratic equation:
4x2 – 5x – 1 = 0
The roots of quadratic equation x(x + 8) + 12 = 0 are ______.
