Quadratic equation
A quadratic equation is an equation in the form of Template:Math, where a is not equal to 0. It makes a parabola (a "U" shape) when graphed on a coordinate plane.
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:
- [math]\displaystyle{ x = \frac{-b \pm \sqrt {b^2-4ac}}{2a} }[/math]
Where the letters are the corresponding numbers of the original equation, Template:Math. Also, a cannot be 0 for the formula to work properly.
The factored form of this equation is Template:Math, where s and t are the zeros, a is a constant, and y and the two xs are ordered pairs which satisfy the equation.
Proof
The quadratic formula is proved by completing the square,
Divide the quadratic equation by a :
- [math]\displaystyle{ x^2 + \frac{b}{a} x + \frac{c}{a}=0,\,\! }[/math]
Move Template:Math:
- [math]\displaystyle{ x^2 + \frac{b}{a} x= -\frac{c}{a}.\,\! }[/math]
Use the method of completing the square
- To "complete the square" is to find some "k" so that:
- [math]\displaystyle{ x^2 + \frac{b}{a} x +k = x^2+2xy+y^2,\,\! }[/math]
- for some y.
- [math]\displaystyle{ y = \frac{b}{2a}\,\! }[/math]
- and
- [math]\displaystyle{ k = y^2,\,\! }[/math]
- so
- [math]\displaystyle{ k = \frac{b^2}{4a^2}.\,\! }[/math]
Add [math]\displaystyle{ k = \frac{b^2}{4a^2}\,\! }[/math] to both sides of the equation:
- [math]\displaystyle{ x^2 + \frac{b}{a} x= -\frac{c}{a},\,\! }[/math]
Which gives:
- [math]\displaystyle{ x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=-\frac{c}{a}+\frac{b^2}{4a^2}.\,\! }[/math]
The left side is now a perfect square; it is the square of
- [math]\displaystyle{ x + \frac{b}{2a}.\,\! }[/math]
The right side can be a single fraction, with a common denominator Template:Math.
- [math]\displaystyle{ \left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}. }[/math]
Find the square root of both sides.
- [math]\displaystyle{ x+\frac{b}{2a}=\pm\frac{\sqrt{b^2-4ac\ }}{2a}. }[/math]
Move Template:Math:
- [math]\displaystyle{ x=-\frac{b}{2a}\pm\frac{\sqrt{b^2-4ac\ }}{2a}=\frac{-b\pm\sqrt{b^2-4ac\ }}{2a}. }[/math]