Quadratics - a MaS Tutor.

Features of a Quadratic Function

Quadratic
Function

Solving
Quadratics

A proof of
the Quadratic Formula

Roots
of a Quadratic

The Difference
of Two Squares DOTS

Features of a Quadratic

Shape of the Curve

graph showing the two general shapes of a quadratic

A quadratic is either opens up, shaped like a "valley", or opens down, like a "hill", depending on the value of the x-squared coefficient.

Intercepts

p and q are intercepts of the quadratic curve with the x-axis and c is with the y-axis

Vertex

The vertex of a quadratic is the turning point of a quadratic graph is its minimum point (for a "valley"), or its maximum point (for a "hill").

equation

equation TBC

Links between the graph and equation

Links between the graph and equation TBC

Quadratic Roots

If $α$ & $β$ are roots of a quadratic then,

$x = α$ & $x = β$

so,

$x - α = 0$ & $x - β = 0$

Multiplying these two to get a quadratic,

(x - α) (x - β) = 0

Multiplying out the brackets and simplifying,

x^{2} - (α + β) x + α β = 0

Scaling by $γ$ ,

γ x^{2} - γ (α + β) x + α β γ = 0

Then comparing coefficients with,

a x^{2} + b x + c = 0

We find,

$a = γ$ , $b = - γ (α + β)$ , & $c = α β γ$ .

Quadratic Function

The formula for a general quadratic is given by

y = a x^{2} + b x + c

where
$y$ is the dependent variable;
$x$ is the independent variable,
&
$a$ , $b$ & $c$ are real valued constants.

graph of simple quadratic curve passing through the origin

Quadratic Inequalities

Parabola

Linking into Conic Sections

Conics

(from www.khanacademy.org)

Types of Quadratic

A quadratic expression is functions of the form ...

a x^{2} + b x + c

where $a$ , $b$ and $c$ are constants.

or, factorised ...

(p x + q) (r x + s)

which is a factorised version of the first, where $p$ , $q$ , $r$ and $s$ are constants.

or, completed square form ...

α {(x + β)}^{2} + γ

again another version of the first, where $α$ , $β$ and $γ$ are constants.

Fully factorise the following:

(Hint: always look for a common factor first, if available, then check if other forms of factorisation are possible)

$x^{2} + 8 x$
$x^{2} + 4 x + 4$
$x^{2} + 4 x - 12$
$x^{2} + 4 x + 5$
$x^{4} - 4 x^{3}$
$x^{2} - 12 x + 35$
$z^{2} - z - 72$
$y^{2} + 2 y - 120$
$m^{2} - 144$
$p^{3} + 5 p^{2} - 36 p$
$q^{4} - 16 q^{2} + 64$
$p^{2} q^{2} - 6 p q + 9$

Challenge

How many integer values of $a$ are there for which $x^{2} + a x + 100$ is factorisable? How about $x^{2} + a x - 100$ ?

Answers:

$x (x + 8)$

Difference of Two Squares

a^{2} - b^{2} = (a + b) (a - b)

Prerequisite materials:

Multiplying out brackets

Factorising quadratics

Proof:

(a + b) (a - b)

= a^{2} - a b + a b - b^{2}

= a^{2} - b^{2}

Example problems:

Example A:

Factorise

x^{2} - 4

= x^{2} - 2^{2}

= (x + 2) (x - 2)

Example B:

Factorise

16 x^{2} - 49 y^{2}

= {(4 x)}^{2} - {(7 y)}^{2}

= (4 x + 7 y) (4 x - 7 y)

Example C:

Find the value of

= 101^{2} - 99^{2}

= (101 + 99) (101 - 99)

= 200 \times 2

= 400

Questions - Expand the brackets:

$(x - y) (x + y)$
$(m + n) (m - n)$
$(p + q) (q - p)$
$(3 x + 5) (3 x - 5)$

- Factorise the following:

$51^{2} - 49^{2}$
$x^{2} - 16$
$25 - x^{2}$
$x^{2} - 25 y^{2}$
$9 x^{2} - 1$
$9 m^{2} - 25$
$64 p^{2} - 121 q^{2}$
$16 x^{4} - 81 y^{4}$

Revision-and Worksheets
(Care of mmerevise.co.uk)

Factorisation: difference of 2 squares Video (DOTS):: Practice Questions;; Textbook Exercise;; Worksheet.
(Care of corbettmaths.com)

Past Paper Questions
(Care of www.goteachmaths.co.uk)

... more questions ...
(Care of duckduckgo.com)

Further study and reading:

Difference of two cubes;
$\cos^{2} θ - \sin^{2} θ$ ;
Sum of two squares.

Quadratic Formula Proof

a x^{2} + b x + c = 0

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

{(x + \frac{b}{2 a})}^{2} - {(\frac{b}{2 a})}^{2} + \frac{c}{a} = 0

{(x + \frac{b}{2 a})}^{2} - \frac{b^{2}}{4 a^{2}} + \frac{c}{a} = 0

{(x + \frac{b}{2 a})}^{2} = \frac{b^{2}}{4 a^{2}} - \frac{c}{a}

{(x + \frac{b}{2 a})}^{2} = \frac{b^{2} - 4 a c}{4 a^{2}}

x + \frac{b}{2 a} = \pm \sqrt{\frac{b^{2} - 4 a c}{4 a^{2}}}

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

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

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