See also: Quadratic Equation, Completing Square, numbers


Given a quadratic equation of the following form,

ax2 + bx + c = 0

the roots can be found using the formula below.

x = -b ± b2 - 4ac 2 a

Using the above formula is probably the easiest and most straightforward way to solve or find the roots of a quadratic equation.



Examples:

  1. Find the roots of the quadratic equation

    0 = x2 4x + 3

    We subsitute the values of the coefficients, a, b and c to the formula to find x.

    In this case, a=1, b=4 and c=3. So

    x = - (-4) ± (-4) 2 - 413 2 1 x = 4 ± 16 - 12 2 x = 4 ± 4 2 x = 4 ± 2 2 x1 = 4 - 2 2 = 1 x2 = 4 + 2 2 = 3
  2. Find the roots of the quadratic equation

    0 = x2 6x + 9

    Similar to the first example, we subsitute the values of the coefficients, a, b and c to the formula to find x.

    In this case, a=1, b=6 and c=9. So

    x = - (-6) ± (-6) 2 - 419 2 1 x = 6 ± 36 - 36 2 x = 6 ± 0 2 x = 6 ± 0 2 x1,2 = 6 2 x1,2 = 3

    This quadratic equation has only 1 root as expected because the discriminant ( b2 4ac ) is 0.

  3. Find the roots of the quadratic equation

    0 = 2 x2 + 2x + 5

    Similar to the above examples, we subsitute the values of the coefficients, a, b and c to the formula to find x.

    In this case, a=2, b=2 and c=5. So

    x = -2 ± 22 - 425 2 2 x = -2 ± 4 - 40 4 x = -2 ± -36 4 x = -2 ± 6 -1 4 x = -2 ± 6 i 4 x = -1 ± 3 i 2 x1 = -1 - 3 i 2 x2 = -1 + 3 i 2

    This quadratic equation has 2 complex roots as expected because the discriminant ( b2 4ac ) is negative.

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By Jimmy Sie

See also: Quadratic Equation, Completing Square, numbers