Quadratic Equations

Algebra -> Algebra -> Quadratic Equations and Parabolas -> Quadratic Equations Lessons -> Quadratic Equation Lesson -> Quadratic Equations (Log On)

Ad: You enter your algebra equation or inequality - Algebrator solves it step-by-step while providing clear explanations. Free on-line demo .
Ad: Algebra Solved!™: algebra software that solves YOUR algebra homework problems with step-by-step help!

Quadratics: solvers

Practice!

Problems, free tutors

Lessons

Word Problems

In Depth

Proof of the quadratic formula

Consider equation ax²+bx+c = 0. What we want to do is complete the square, that is, get an equation like this:

(...)² = ...

Divide both parts by a:

x^2 + (b/a)x + c/a = 0

x^2 + 2(b/2a)x + c/a = 0

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

Now, if we want to extract a square of a binomial out of here, we can add (b/2a)^2 to both sides of the equation:

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

We did it so that the formula to the left would be a complete square of an expression:

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

Slow down here for just one second. This place is crucial for understanding the discriminant. On the left side, there is a square of something, on the right, it is a number. As you know, for all real numbers their squares are non-negative. So if the number to the right happens to be negative, this means that there is no real value of x that would satisfy this equation. That's where the discriminant rule comes from.

Since the roots of equations like y² = z are y=0 +-sqrt(z) , we have: