If the roots of the equations a x 2 + 2 b x + c = 0 and b x 2 - 2 a c x + b = 0 are simultaneously real, then prove that b 2 = a c .

We have, a x 2 + 2 b x + c = 0 . . . . . . . . ( i ) and, b x 2 - 2 a c x + b = 0 . . . . . . . . . . ( ii ) Let D 1 and D 2 be the discriminant of equations ( i ) and ( ii ) respectively. Then, D 1 = ( 2 b ) 2 - 4 a c = 4 b 2 - 4 a c ⇒ D 1 = 4 b 2 - 4 a c ≥ 0 [ ∵ roots are real] and D 2 = ( - 2 a c ) 2 - 4 b 2 = 4 a c - 4 b 2 ⇒ D 2 = 4 a c - 4 b 2 ≥ 0 [ ∵ roots are real] Now, D 1 ≥ 0 ⇒ 4 b 2 - 4 a c ≥ 0 ⇒ 4 b 2 ≥ 4 a c ⇒ b 2 ≥ a c . . . . . . . . . . ( iii ) and, D 2 ≥ 0 ⇒ 4 a c - 4 b 2 ≥ 0 ⇒ 4 a c ≥ 4 b 2 ⇒ a c ≥ b 2 . . . . . . . . ( iv ) Using equations ( iii ) and ( iv ) , we get, b 2 = a c

If a , b , c are real numbers such that a c ≠ 0 , then show that at least one of the equations a x 2 + b x + c = 0 and - a x 2 + b x + c = 0 has real roots.

We have, a x 2 + b x + c = 0 . . . . . . . . ( i ) and - a x 2 + b x + c = 0 . . . . . . ( ii ) The discriminant of equation ( i ) is given by D 1 = b 2 - 4 a c Now, the discriminant of equation ( ii ) is given by D 2 = b 2 - 4 ( - a ) c ⇒ D 2 = b 2 + 4 a c Now, D 1 + D 2 = b 2 - 4 a c + b 2 + 4 a c ⇒ D 1 + D 2 = 2 b 2 ≥ 0 [ ∴ b is real] ⇒ At least one of D 1 and D 2 is greater than or equal to zero. Hence, at least one of the two equations has real roots.

E M B I B E

MATHEMATICS CLASS X>Chapter 4 - Quadratic Equations>EXERCISE 4.6

R. D. Sharma Solutions for Chapter: Quadratic Equations, Exercise 6: EXERCISE 4.6

Author:R. D. Sharma

R. D. Sharma Mathematics Solutions for Exercise - R. D. Sharma Solutions for Chapter: Quadratic Equations, Exercise 6: EXERCISE 4.6

Attempt the free practice questions on Chapter 4: Quadratic Equations, Exercise 6: EXERCISE 4.6 with hints and solutions to strengthen your understanding. MATHEMATICS CLASS X solutions are prepared by Experienced Embibe Experts.

Questions from R. D. Sharma Solutions for Chapter: Quadratic Equations, Exercise 6: EXERCISE 4.6 with Hints & Solutions

MEDIUM

10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 4 - Quadratic Equations>EXERCISE 4.6>Q 15

Determine the nature of the roots of the following quadratic equation:

$2 (a^{2} + b^{2}) x^{2} + 2 (a + b) x + 1 = 0$

MEDIUM

10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 4 - Quadratic Equations>EXERCISE 4.6>Q 18

If the roots of the equation $(a^{2} + b^{2}) x^{2} - 2 (a c + b d) x + (c^{2} + d^{2}) = 0$ are equal, prove that $\frac{a}{b} = \frac{c}{d}$ .

MEDIUM

10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 4 - Quadratic Equations>EXERCISE 4.6>Q 19

If the roots of the equations $a x^{2} + 2 b x + c = 0$ and $b x^{2} - 2 \sqrt{a c} x + b = 0$ are simultaneously real, then prove that $b^{2} = a c$ .

MEDIUM

10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 4 - Quadratic Equations>EXERCISE 4.6>Q 20

If $p, q$ are real and $p \neq q$ , then show that the roots of the equation $(p - q) x^{2} + 5 (p + q) x - 2 (p - q) = 0$ are real and unequal.

MEDIUM

10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 4 - Quadratic Equations>EXERCISE 4.6>Q 21

If the roots of the equation $(c^{2} - a b) x^{2} - 2 (a^{2} - b c) x + b^{2} - a c = 0$ are equal, prove that either $a = 0$ or $a^{3} + b^{3} + c^{3} = 3 a b c$ .

MEDIUM

10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 4 - Quadratic Equations>EXERCISE 4.6>Q 23

Prove that both the roots of the equation $(x - a) (x - b) + (x - b) (x - c) + (x - c) (x - a) = 0$ are real but they are equal only when $a = b = c$ .

MEDIUM

10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 4 - Quadratic Equations>EXERCISE 4.6>Q 24

If $a, b, c$ are real numbers such that $a c \neq 0$ , then show that at least one of the equations $a x^{2} + b x + c = 0$ and $- a x^{2} + b x + c = 0$ has real roots.