Find the roots of the following equation x - 1 x = 3 , x ≠ 0 .

Given: x - 1 x = 3 ⇒ x 2 - 3 x - 1 = 0 We use the quadratic formula to evaluate the roots x = - b ± b 2 - 4 a c 2 a ⇒ x = - ( - 3 ) ± ( - 3 ) 2 - 4 × 1 × - 1 2 × 1 ⇒ x = 3 ± 13 2 The two roots are 3 + 13 2 , 3 - 13 2

Find the roots of the following equation x - 1 x = 3 , x ≠ 0 .

Given: x - 1 x = 3 ⇒ x 2 - 3 x - 1 = 0 We use the quadratic formula to evaluate the roots x = - b ± b 2 - 4 a c 2 a ⇒ x = - ( - 3 ) ± ( - 3 ) 2 - 4 × 1 × - 1 2 × 1 ⇒ x = 3 ± 13 2 The two roots are 3 + 13 2 , 3 - 13 2

Find the roots of the following equation x + 1 x = 3 , x ≠ 0

Given x + 1 x = 3 On squaring to both side ⇒ x 2 - 3 x + 1 = 0 The roots of a quadratic equation a x 2 + b x + c given by x = - b ± b 2 - 4 a c 2 a Therefore, the required roots are x = 3 ± 5 2 ⇒ x = 3 + 5 2 and 3 - 5 2

Solve the following equation by formula method. x 2 - 5 x - 6 = 0 .

Given: x 2 - 5 x - 6 = 0 The formula for finding roots of a quadratic equation a x 2 + b x + c = 0 is x = - b ± b 2 - 4 a c 2 a . For the equation x 2 - 5 x - 6 = 0 , a = 1 , b = - 5 , and c = - 6 . ∴ x = - - 5 ± - 5 2 - 4 × 1 × - 6 2 × 1 ⇒ x = 5 ± 25 + 24 2 ⇒ x = 5 ± 49 2 ⇒ x = 5 ± 7 2 ⇒ x = 5 + 7 2 = 6 or x = 5 - 7 2 = - 1 ⇒ x = 6 or - 1 Therefore, the solutions of x 2 - 5 x - 6 = 0 are 6 and - 1 .

Write the condition of a quadratic equation a x 2 + b x + c = 0 has no real root

Nature of the quadratic equation depends on the discriminant of a quadratic equation. We know that, a quadratic equation a x 2 + b x + c = 0 has no real root when the discriminant is negative. D < 0 i.e, b 2 - 4 a c < 0