Quadratic equation

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A quadratic equation graphed in the coordinate plane.
A quadratic equation graphed in the coordinate plane.

A quadratic equation is an equation in the form of ax2 + bx + c, where a is not equal to 0. It makes a parabola (a "u" shape) when graphed on a coordinate plane.

[edit] The Quadratic Formula

The quadratic formula is a formula used to find the points where the graphed equation crosses the x-axis, or the horizontal axis. These points are called the "zeroes" of a function. The formula is:

x = \frac{-b \pm \sqrt {b^2-4ac}}{2a}

Where the letters are the corresponding numbers of the original equation, ax2 + bx + c. a is not 0

[edit] Proof

The quadratic formula is prooved by completing the square,

Divide the quadratic equation by a :

x^2 + \frac{b}{a}  x + \frac{c}{a}=0,\,\!

move \frac{c}{a}.\,

x^2 + \frac{b}{a} x= -\frac{c}{a}.\,\!

Use the method of completing the square

To "complete the square" is to find some "k" so that:
x^2 + \frac{b}{a} x +k = x^2+2xy+y^2,\,\!
for some y.
y = \frac{b}{2a}\,\!
and
k = y^2,\,\!
so
k = \frac{b^2}{4a^2}.\,\!


Add k = \frac{b^2}{4a^2}\,\! to both sides of the equation:


x^2 + \frac{b}{a} x= -\frac{c}{a},\,\!

makes:

x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=-\frac{c}{a}+\frac{b^2}{4a^2}.\,\!

The left side is now a perfect square; it is the square of

x + \frac{b}{2a}.\,\!

The right side can be a single fraction, with a common denominator 4a2.

\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}.

Find the square root of both sides.

x+\frac{b}{2a}=\pm\frac{\sqrt{b^2-4ac\  }}{2a}.

move \frac{b}{2a}

x=-\frac{b}{2a}\pm\frac{\sqrt{b^2-4ac\  }}{2a}=\frac{-b\pm\sqrt{b^2-4ac\  }}{2a}.