How do you solve a quadratic equation?

Answer 1

There are many ways to solve a quadratic equation, but the quadratic formula works for all equations and is very quick. The formula is

x= -b +/- the square root of (b^2 - 4ac)

_________________________________

2a

To find a,b, and c refer to the layout of a quadratic equation:

ax2 + bx +c

New Answer (from Nghi1350).

If the given quadratic equation can be factored, you can solve it faster by using the factoring "ac method" (You Tube) or by the new Diagonal Sum method (Amazon e-book 2010).

Otherwise, use the quadratic formula. There is an improved quadratic formula that is easier to remember presented in the above mentioned book. This formula is called the "Quadratic formula in graphic form", since it relates the real roots to the x-intercepts of the parabola graph of the quadratic function.

The 2 real roots are given by this formula:

x1 = - b/2a + d/2a ; and x2 = -b/2a - d/2a. (1)

The quantity (-b/2a) represents the x-coordinate of the symmetry axis of the parabola.

The 2 quantities (d/2a) and (-d/2a) represent the 2 distances from this axis to the two x-intercepts of the parabola.

The quantity (d) can be zero, a number, or imaginary.

- If d = 0; there is double root at x = -b/2a

- If d is a number (real or radical): there are 2 real roots.

- If d is imaginary: There are no real roots.

The quantity (d) is given by the relation (2), obtained by writing that the product of the 2 real roots is equal to (c/a):

[(-b - d)/2a][-b + d)/2a] = c/a

b^2 - d^2 = 4ac

d^2 = b^2 - 4ac (2)

To solve a quadratic equation, first find d by the relation (2) then find the real roots by the formula (1).

This new improved quadratic formula is easier to remember since you can relate it to the x-intercepts of the parabola graph. In addition, the quantity (d/2a) makes more sense about distance than the classical quantity "square root of b^2 - 4ac".