Quadratic equation

A quadratic equation can take two forms. The general formula, written as a function of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , $y=f(x)$, is:

$y=ax^{2}+bx+c$

The graph of a quadratic equation is a parabola, one of the conic sections. (In some cases, the parabola collapses, most obviously when $a=0$)

The points where this curve crosses the x axis are represented by the second form of the equation:

$ax^{2}+bx+c=0$

These are solved for using the quadratic formula, which will not only solve for real roots, but result in the imaginary roots if the parabola does not actually cross the y axis (this is when $4ac$ is greater than $b^{2}$).

Quadratic equations are very important in calculating the motion of bodies under constant acceleration, i.e., gravity (when close to the earth's surface).

The derivative of a quadratic equation is a simple linear function:

${\frac {dy}{dx}}=2ax+b$

Factoring

Quadratic equations can be simplified by factoring it into $(x+r)(x+s)$, where $r+s$ equals B in $y=x^{2}+bx+c$ and $rs$ equals Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} in the above equation. Equations that cannot be easily factored this way can become easy to factor by completing the square.

If the coefficient of X is not one

If the coefficient of X is not 1, then one can turn Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax^2 + bx} into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a(x^2 + b/ax)} . This works because both Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle bx} were divided by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , which was put back later via multiplication.