Root nature

To find the solution of a quadratic equation ax+ bx + c = 0 we use the quadratic formula, which has the following form:

x = [-b ± √ (b- 4ac)] / 2a

Substituting the values of the coefficients a, b and c into it we can easily obtain the two roots of x, which satisfy this equation.

Part "b- 4ac = D ”is called discriminating and depending on its value, we will know the nature of the roots from the quadratic equation:

  • If D> 0, the roots of that equation will be real and different.
  • If D = 0, the roots of the equation will be real and equal.
  • If D <0, the roots will be complex.

Examples:

1. Find the nature of the roots of the following equation: 20x² - x - 1 = 0.

The coefficients are: a = 20; b = - 1 and c = - 1. By replacing them in the discriminant D = b- 4ac, we have:

  D = (- 1)- 4 (20) (- 1) = 1 + 80 = 81.

Since D> 0, the roots will be real and different.

2. Find the nature of the roots of the following equation: 4x² - 16x + 16 = 0.

The coefficients are: a = 4; b = - 16 and c = 16. When replacing them in the discriminant D = b- 4ac, we have:

  D = (- 16)- 4 (4) (16) = 256 - 256 = 0.

Since D = 0, the roots will be real and equal.

3. Find the nature of the roots of the following equation: 4x² - 8x + 7 = 0.

The coefficients are: a = 4; b = - 8 and c = 7. By replacing them in the discriminant D = b- 4ac, we have:

  D = (- 8)- 4 (4) (7) = 64 - 112 = - 48.

Since D <0, the roots will be complex.