Quadratic equation

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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.

The Quadratic Formula[change | edit source]

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 = 0. Also, a cannot be 0 for the formula to work properly.

The factored form of this equation is y = a(xs)(xt), 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[change | edit source]

The quadratic formula is proved by completing the square,

Divide the quadratic equation by a :

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

Move 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},\,\!

Which gives:

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 b/2a:

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

Other websites[change | edit source]