Expressing the left side as a perfect square,

( x + b/(2a) )^2 = (b/(2a))^2 - c/a = (b^2 - 4ac) / (4a^2)

Solving the square root of the left side,

x + b/(2a) = ±((b^2 - 4ac) / (4a^2))^(1/2)

Thus,

x = -(b/(2a))±((b^2 - 4ac) / (4a^2))^(1/2) 

Arranging the above, we get x as below :

x = (-b±(b^2 - 4ac)^(1/2)) / (2a)

where, letting D = (b^2 - 4ac) (D is generally called as the discriminant.),

If D > 0, x has two answers.
If D = 0, x has only one answer.
If D < 0, x has no answer . (imaginary root)


For the general form of quadratic equations ( y = ax^2 + bx + c, a≠0 ), 

Answers which are the intercept to the x axis (i.e. y=0) can be obtained by the above description.

If a > 0, the function is convex parabolic and has a minimum value.


If a < 0, the function is concave parabolic and has a maximum value.

Value of y as a quadratic function is calculated as follows by the
value of x,

Discriminant D is :

If D >= 0, answers x1and x2 are :

Also, expressing into a perfect square,

y = a(x+b/(2a))^2 - (b^2 - 4ac)/(4a)

Then, the coordinates of the vertex are :

xp = -b/(2a), yq = - (b^2 - 4ac)/(4a)