NATURE OF ROOTS OF QUADRATIC EQUATION DISCRIMINANT

The roots of the quadratic equation ax² +bx +c = 0 , a ≠ 0 are found using the formula x = [-b ± √(b² - 4ac)]/2a

Here, b² - 4ac called as the discriminant (which is denoted by D ) of the quadratic equation, decides the nature of roots as follows

Question :

Determine the nature of the roots for the following quadratic equations

(i) 15x² + 11x + 2 = 0

Solution :

By comparing the given quadratic equation with the general form of quadratic equation.

ax² + bx + c = 0

a = 15, b = 11 and c = 2

Δ = b² - 4ac

Δ = 11² - 4(15)(2)

  Δ  =  121 - 120

  Δ   =  1 > 0

Hence the roots are real and unequal.

(ii) x² − x − 1 = 0

Solution :

a = 1, b = -1 and c = -1

Δ = b² - 4ac

Δ = (-1)² - 4(1)(-1)

  Δ  =  1 + 4

  Δ   =  5 > 0

Hence the roots are real and unequal.

(iii)  √2t² −3t + 3√2 = 0

Solution :

a = √2, b = -3 and c = 3√2

Δ = b² - 4ac

Δ = (-3)² - 4(√2)(3√2)

  Δ  =  9 - 12(2)

  Δ  =  9 - 24

  Δ   =  -15 < 0

Hence it has no real roots.

(iv)  9y² − 6√2 y + 2 = 0

Solution :

a = 9, b = − 6√2 and c = 2

Δ = b² - 4ac

Δ = (− 6√2)² - 4(9)(2)

  Δ  =  36(2) - 72

  Δ  =  72 - 72

  Δ   =  0

Hence it has no real and equal roots.

(v)  9a²b²x² −24abcdx + 16c²d² = 0 , a ≠ 0 , b ≠ 0

Solution :

a = 9a²b², b = −24abcd and c = 16c²d²

Δ = b² - 4ac

Δ = ( −24abcd)² - 4(9a²b²)(16c²d²)

  Δ  =  576 a²b²c²d² - 576 a²b²c²d²

  Δ  =  0

Hence it has no real and equal roots.

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