Notes

== Quadratic equation subject ==

=== Steps ===

''ax''² + ''bx'' + ''c'' = 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''

2''ax'' + ''b'' = ±{{sqrt|''b''² – 4''ac''}}

2''ax'' = –''b'' ± {{sqrt|''b''² – 4''ac''}}

''x'' = {{sfrac|–''b'' ± {{sqrt|''b''² – 4''ac''}}|2''a''}}

=== How to make ''x'' the quadratic equation subject, i.e. derive the quadratic formula ===

* Multiply the left side by 4''a'', as the right side is zero.

* Add –4''ac'' to both sides.

* Add ''b''² to both sides to complete the square.

* Square-root both sides.

* Add –''b'' to both sides.

* Divide both sides by 2''a''.

=== How to turn the quadratic equation, written as ''ax''² + ''bx'' + ''c'' = 0, into (2''ax'' + ''b'')² = ''b''² – 4''ac'' ===

* Replace 4''ac'' by ''b''² on the left side.

* Keep (''b''² – 4''ac'') on the right side.

=== ''b''² – 4''ac'' ===

* (''b''² – 4''ac'') is the expression that determines how many solutions quadratic equations, written as ''ax''² + ''bx'' + ''c'' = 0, have.

* When ''b''² – 4''ac'' > 0, there are two solutions.

* When ''b''² – 4''ac'' = 0, the only solution is ''x'' = –{{sfrac|''b''|2''a''}}.

* When ''b''² – 4''ac'' < 0, there are no solutions.

* When (''b''² – 4''ac'') is a perfect square, there are two rational solutions.

* When (''b''² – 4''ac'') is not a perfect square, there are two irrational solutions.

== Quadratic expression forms ==

=== Vertex form conversion ===

''ax''² + ''bx'' + ''c'' = {{sfrac|(2''ax'' + ''b'')² + 4''ac'' – ''b''²|4''a''}}

=== Quadratic expression forms ===

* Quadratic expressions, written as (''ax''² + ''bx'' + ''c''), become {{sfrac|(2''ax'' + ''b'')² + 4''ac'' – ''b''²|4''a''}} when converted to their vertex forms.

* Quadratic expressions, written as (''ax''² + ''bx'' + ''c''), can be factorized when (''b''² – 4''ac'') is a perfect square.

* When ''a'' > 0, quadratic expressions have minimum values, i.e. ''ax''² + ''bx'' + ''c'' >= {{sfrac|(4''ac'' – ''b''²|4''a''}}.

* When ''a'' < 0, quadratic expressions have maximum values, i.e. ''ax''² + ''bx'' + ''c'' <= {{sfrac|(4''ac'' – ''b''²|4''a''}}.

== Parabolae ==

=== Regular form ===

''y'' = ''ax''² + ''bx'' + ''c''

=== ''x''-intercept count ===

* When ''b''² > 4''ac'', there are two ''x''-intercepts, i.e. parabolae cross or cut the ''x''-axis twice.

* When ''b''² = 4''ac'', the only ''x''-intercept is ''x'' = –{{sfrac|''b''|2''a''}}, i.e. parabolae touch or hit the ''x''-axis only once.

* When ''b''² < 4''ac'', there are no ''x''-intercepts, i.e. parabolae never intersect the ''x''-axis.

=== Vertex figures ===

* (–{{sfrac|''b''|2''a''}}, {{sfrac|(4''ac'' – ''b''²|4''a''}}) is the vertex figure lying on the parabola, written as ''y'' = ''ax''² + ''bx'' + ''c''

* When ''a'' > 0, parabolae have minimum vertex figures.

* When ''a'' < 0, parabolae have maximum vertex figures.

== Theories focusing on quadratic expression coefficients ==

''a'' > 0, ''b'' < 0, ''c'' > 0, ''b''² > 4''ac''

''a'' > 0, ''b'' < 0, 0 < ''c'' < {{sfrac|''b''²|4''a''}}

''a'' > 0, ''b'' < –2{{sqrt|''ac''}}, ''c'' > 0

0 < ''a'' < {{sfrac|''b''²|4''c''}}, ''b'' < 0, ''c'' > 0

''a'' > 0, ''b'' < –2{{sqrt|''ac''}}, 0 < ''c'' < {{sfrac|''b''²|4''a''}}

0 < ''a'' < {{sfrac|''b''²|4''c''}}, ''b'' < 0, 0 < ''c'' < {{sfrac|''b''²|4''a''}}

0 < ''a'' < {{sfrac|''b''²|4''c''}}, ''b'' < –2{{sqrt|''ac''}}, ''c'' > 0

0 < ''a'' < {{sfrac|''b''²|4''c''}}, ''b'' < –2{{sqrt|''ac''}}, 0 < ''c'' < {{sfrac|''b''²|4''a''}}

== Quadratic function switch ==

=== Steps ===

''ax''² + ''bx'' + ''c'' = ''y''

4''a''²''x''² + 4''abx'' + 4''ac'' = 4''ay''

4''a''²''x''² + 4''abx'' = 4''a''(''y'' – ''c'')

4''a''²''x''² + 4''abx'' + ''b''² = ''b''² + 4''a''(''y'' – ''c'')

(2''ax'' + ''b'')² = ''b''² + 4''a''(''y'' – ''c'')

2''ax'' + ''b'' = ±{{sqrt|''b''² + 4''a''(''y'' – ''c'')}}

2''ax'' = –''b'' ± {{sqrt|''b''² + 4''a''(''y'' – ''c'')}}

''x'' = {{sfrac|–''b'' ± {{sqrt|''b''² + 4''a''(''y'' – ''c'')}}|2''a''}}

== Tangents to the parabola at intersection points relative to the horizontal axis ==

Given ''y'' = ''ax''² + ''bx'' + ''c'' where ''a'' > 0, ''b'' < 0, ''c'' > 0 and ''b''² > 4''ac''

(2''ax'' + ''b''){{sqrt|''b''² – 4''ac''}} + 2''ay'' = 4''ac'' – ''b''² at (–{{sfrac|''b'' + {{sqrt|''b''² – 4''ac''}}|2''a''}}, 0)

2''ay'' = (2''ax'' + ''b''){{sqrt|''b''² – 4''ac''}} + 4''ac'' – ''b''² at ({{sfrac|–''b'' + {{sqrt|''b''² – 4''ac''}}|2''a''}}, 0)

Tangents to the parabola, written as ''y'' = ''ax''² + ''bx'' + ''c'', at intersection points relative to the ''x''-axis where ''a'' > 0, ''b'' < 0, ''c'' > 0 and ''b''² > 4''ac''