Quadratic equation

A quadratic equation is a polynomial equation of a single variable in which the highest exponent of the variable is two. The following is a quadratic equation expressed in the standard form:

$$y=a{x}^2+bx+c$$

where:

• $a$ is the quadratic coefficient, which controls the “width” and shape of the parabola ($a<0⟹$ sad face, $a>0⟹$ smiley face)
• $b$ is the linear coefficient, which controls the horizontal position and slope at the $y$-intercept
• $c$ is the constant coefficient, which specifies the $y$-intercept

The equation is said to have a solution if $a{x}^2+bx+c=0$. A quadratic equation may have up to two solutions, and they can easily be determined by calculating the discriminant of the equation:

$$b^2-4ac$$

Discriminant	Suggestion
$b^2-4ac>0$	2 real solutions
$b^2-4ac=0$	1 real solution
$b^2-4ac<0$	No real solutions (2 complex solutions)

Forms

A quadratic equation can be manipulated into different forms depending on the operations performed on it.

Standard	Factored	Vertex
$y=a{x}^2+bx+c$ $y$-intercept	$y=a(x-p)(x-q)$ $x$-intercepts at $(p,0)$ and $(q,0)$	$y=a(x-h{)}^2+k$ Vertex (minimum/maximum) point at $(h,k)$

Solving

Quadratic equations may be solved using three methods:

• Factoring the quadratic
• Completing the square
• Using the quadratic formula