Advertisements
Advertisements
प्रश्न
If p, q are real and p ≠ q, then show that the roots of the equation (p − q) x2 + 5(p + q) x− 2(p − q) = 0 are real and unequal.
Advertisements
उत्तर
The quadric equation is (p − q) x2 + 5(p + q) x− 2(p − q) = 0
Here,
a = (p - q), b = 5(p + q) and c = -2(p - q)
As we know that D = b2 - 4ac
Putting the value of a = (p - q), b = 5(p + q) and c = -2(p - q)
D = {5(p + q)}2 - 4 x (p - q) x (-2(p - q))
= 25(p2 + 2pq + q2) + 8(p2 - 2pq + q2)
= 25p2 + 50pq + 25q2 + 8p2 - 16pq + 8q2
= 33p2 + 34pq + 33q2
Since, P and q are real and p ≠ q, therefore, the value of D ≥ 0.
Thus, the roots of the given equation are real and unequal.
Hence, proved
APPEARS IN
संबंधित प्रश्न
If the quadratic equation px2 − 2√5px + 15 = 0 has two equal roots then find the value of p.
If x=−`1/2`, is a solution of the quadratic equation 3x2+2kx−3=0, find the value of k
Determine the nature of the roots of the following quadratic equation:
`3/5x^2-2/3x+1=0`
Find the values of k for which the roots are real and equal in each of the following equation:
`kx^2-2sqrt5x+4=0`
Determine the nature of the roots of the following quadratic equation :
4x2 - 8x + 5 = 0
Solve the following quadratic equation using formula method only :
16x2 = 24x + 1
Solve the following quadratic equation using formula method only :
x2 +10x- 8= 0
If one root of the equation 2x² – px + 4 = 0 is 2, find the other root. Also find the value of p.
Find the discriminant of the following equations and hence find the nature of roots: 3x2 – 5x – 2 = 0
Without solving the following quadratic equation, find the value of ‘p’ for which the given equations have real and equal roots: x2 + (p – 3)x + p = 0.
Find the value(s) of k for which each of the following quadratic equation has equal roots: (k + 4)x2 + (k + 1)x + 1 =0 Also, find the roots for that value (s) of k in each case.
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.
x2 – 3x + 4 = 0
State whether the following quadratic equation have two distinct real roots. Justify your answer.
`(x - sqrt(2))^2 - 2(x + 1) = 0`
Compare the quadratic equation `x^2 + 9sqrt(3)x + 24 = 0` to ax2 + bx + c = 0 and find the value of discriminant and hence write the nature of the roots.
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 ______.
The roots of equation (q – r)x2 + (r – p)x + (p – q) = 0 are equal.
Prove that 2q = p + r; i.e., p, q, and r are in A.P.
Which of the following equations has two real and distinct roots?
One root of equation 3x2 – mx + 4 = 0 is 1, the value of m is ______.
