RS Aggarwal - Mathematics

Book: RS Aggarwal - Mathematics

Chapter: 10. Quadratic Equations

Subject: Maths - Class 10th

Q. No. 15 of Exercise 10D

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15

If the roots of the equation (c2 – ab)x2 – 2(a2 – bc)x + (b2 – ac) = 0 are real and equal, show that either a = 0 or (a3 + b3 + c3) = 3abc.

Given that the roots of the equation (c2 – ab)x2 – 2(a2 – bc)x + (b2 – ac) = 0 are real and equal

Comparing with standard quadratic equation ax2 + bx + c = 0


a = (c2 – ab) b = – 2(a2 – bc) c = (b2 – ac)


Thus D = 0


Discriminant D = b2 – 4ac = 0


[ – 2(a2 – bc)]2 – 4(c2 – ab) (b2 – ac) = 0


4(a4 – 2a2bc + b2c2) – 4(b2c2 – ac3 – ab3 + a2bc) = 0


using (a – b)2 = a2 – 2ab + b2


a4 – 2a2bc + b2c2 – b2c2 + ac3 + ab3 – a2bc = 0


a4 – 3a2bc + ac3 + ab3 = 0


a (a3 – 3abc + c3 + b3) = 0


a = 0 or (a3 – 3abc + c3 + b3) = 0


Hence proved a = 0 or a3 + c3 + b3 = 3abc


Chapter Exercises

Exercise 10A
Exercise 10B
Exercise 10C
Exercise 10D
Exercise 10E
Exercise 10F

More Exercise Questions

1

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2x2 – 8x + 5 = 0
1

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3x2 –2√6x + 2 = 0
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5x2 – 4x + 1 = 0
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5x (x – 2) + 6 = 0
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12x2 – 4√15x + 5 = 0
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Find the nature of the roots of the following quadratic equations:

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2

If a and b are distinct real numbers, show that the quadratic equation 2 (a2 + b2)x2 + 2(a + b)x + 1 = 0 has no real roots.
3

Show that the roots of the equation x2 + px – q2 = 0 are real for all real values of p and q.
4

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5

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6

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7

Find the nonzero value of k for which the roots of the quadratic equation 9x2 – 3kx + k = 0 are real and equal.
8

Find the values of k for which the quadratic equation (3k + 1) x2 + 2(k + 1)x + 1 = 0 has real and equal roots.
9

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10

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11

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.
12

If 3 is a root of the quadratic equation x2 – x + k = 0, find the value of p so that the roots of the equation x2 + k (2x + k + 2) + p = 0 are equal.
13

If – 4 is a root of the equation x2 + 2x + 4p = 0, find the value of k for which the quadratic equation x2 + px (1 + 3k) + 7(3 + 2k) = 0 has equal roots.
14

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15

If the roots of the equation (c2 – ab)x2 – 2(a2 – bc)x + (b2 – ac) = 0 are real and equal, show that either a = 0 or (a3 + b3 + c3) = 3abc.
16

Find the values of p for which the quadratic equation 2x2 + px + 8 = 0 has real roots.
17

Find the value of a for which the equation (a – 12)x2 + 2(a – 12)x + 2 = 0 has equal roots.
18

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19

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20

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21

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22

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