English

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
Advertisements

Question

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

Solution

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

shaalaa.com
  Is there an error in this question or solution?
Chapter 4: Quadratic Equations - Exercise 4.6 [Page 43]

APPEARS IN

R.D. Sharma Mathematics [English] Class 10
Chapter 4 Quadratic Equations
Exercise 4.6 | Q 20 | Page 43

RELATED QUESTIONS

Solve the equation by using the formula method. 3y2 +7y + 4 = 0


 

Solve for x: `1/(x+1)+2/(x+2)=4/(x+4), `x ≠ -1, -2, -3

 

If ad ≠ bc, then prove that the equation (a2 + b2) x2 + 2 (ac + bd) x + (c2 + d2) = 0 has no real roots.


Form the quadratic equation if its roots are –3 and 4.


Find the values of k for which the roots are real and equal in each of the following equation:

x2 - 2kx + 7k - 12 = 0


In the following determine the set of values of k for which the given quadratic equation has real roots:

2x2 + kx + 3 = 0


In the following determine the set of values of k for which the given quadratic equation has real roots:

2x2 + kx + 2 = 0


In the following determine the set of values of k for which the given quadratic equation has real roots:

4x2 - 3kx + 1 = 0


Find the value(s) of k so that the quadratic equation 3x2 − 2kx + 12 = 0 has equal roots ?


Find the value of the discriminant in the following quadratic equation: 

x2 +2x-2=0 


Solve the following by reducing them to quadratic equations:
z4 - 10z2 + 9 = 0.


Discuss the nature of the roots of the following quadratic equations : `3x^2 - 2x + (1)/(3)` = 0


If `(1)/(2)` is a root of the equation `x^2 + kx - (5)/(4) = 0`, then the value of k is ______.


If α, β are roots of the equation x2 + 5x + 5 = 0, then equation whose roots are α + 1 and β + 1 is:


The roots of quadratic equation 5x2 – 4x + 5 = 0 are:


Find the roots of the quadratic equation by using the quadratic formula in the following:

–x2 + 7x – 10 = 0


Find whether the following equation have real roots. If real roots exist, find them.

–2x2 + 3x + 2 = 0


If α and β are the distinct roots of the equation `x^2 + (3)^(1/4)x + 3^(1/2)` = 0, then the value of α9612 – 1) + β9612 – 1) is equal to ______.


The roots of the quadratic equation x2 – 6x – 7 = 0 are ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×