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Quadratic Equations Explained

A quadratic equation is an equation that looks like this:

ax 2 + bx + c = 0

where a, b, and c are numbers, called coefficients. For example:

x 2 + 3 x + 4 = 0

You can think about a quadratic equation in terms of a graph of a quadratic function. The equation means that you have to find the points on the horizontal axis x where the graph intersects with the axis.

To solve a quadratic equation, you have to calculate a number called discriminant, usually denoted as d:

d = b 2 4 ac

Depending on the value of d, there are the folowing three possibilities:

  1. Discriminant d is greater than zero. The equation has two solutions:

    x = b ± b 2 - 4 a c 2 a

  2. Discriminant is zero. There is only one solution.

    x = b 2 a

  3. Discriminant is less than zero. No solutions are defined.

Quadratic Derived

4 a ċ ( ax 2 + bx + c ) = 4 a ċ 0 = 0
4 a 2 x 2 + 4 abx + 4 ac = 0
4 a 2 x 2 + 4 abx = 4 ac
4 a 2 x 2 + 4 abx + b 2 = b 2 4 ac
( 2 ax + b ) 2 = b 2 4 ac
4 a 2 x 2 + 4 abx + b 2 = b 2 4 ac
( 2 ax + b ) 2 = b 2 4 ac
2 ax + b = ± b 2 4 ac
x = b ± b 2 4 ac 2 a