Analytical Toolbox

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Analytical Toolbox

• Higher order Equations and Solutions
• By
• Dr J.P.M. Whitty

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Learning objectives

• After the session you will be able to
• Find the solution to Quadratic equations using
  graphical means.
• Find algebraic solutions to quadratic equations.
• Solve simultaneous equations one quadratic and
  one linear.
• Use graphical techniques to estimate the solution
  of higher order equations
• Use math software to solve systems of equations.

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Learning Check Solve simultaneous equations

• Solve the following simultaneous equations
• y2x4 3x7-5y using three different
  mathematical techniques, i.e
• Algebraically
• Graphically
• MATLAB matrix method

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Solution

• -2xy4
• 3x5y7
• -6x3y12
• 6x10y14
• 13y26 y2
• -2x24
• x(4-2)/-2-1

Commands gtgt ezplot('-2xy4') gtgt hold on gtgt
ezplot('3x5y7') gtgt grid on gtgt ginput
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Solution cont

• The MATLAB matrix method is as follows
• A-2 1 3 5
• b4 7
• A\b
• ans-1 2

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Recap Linear Functions

• We have already learned about linear functions,
  for example
• f(x) y 4x 1
• g(x) y -6x -2
• h(x) y (-2/3)x 2

The graphs of these functions are straight lines
and each equation gives us clues as to what
each line looks like f(x) 4x 1
Slope of 4
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Quadratic Functions

• What do the functions
• f(x) 2x2 4x 3
• g(x) x2 4x
  
  h(x) -x2 5x 6
• have in common?

They are all QUADRATIC FUNCTIONS that can be
written in standard form f(x) ax2 bx c
where a ? 0
Pay close attention these coefficients a, b
and c will be clues that help us to know what the
graphs of these quadratic functions look like
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Graphs of Quadratic Functions

• Quadratic Functions graph into a shape called a
  Parabola

If the coefficient of the x2 term, a, is
negative, the parabola opens downward (Sad)
If the coefficient of the x2 term, a, is
positive, the parabola opens upward (Happy)
f(x) -x2 5
Maximum Point
f(x) x2 - 4
Vertex
Minimum Point
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The parabola

• The graph of is a parabola with
  
• vertex (0, 0)

Example Locate the vertex and find the axis of
symmetry of the following parabola. Does it open
up or down? Graph the parabola.
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Quadratic equations

• Any equation that can be written so that one side
  is a quadratic and the other side is zero is
  called a quadratic equation.

In fact, we can consider a general form of
quadratic equation. However a quadratic is given
to us, we can always rewrite it as
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Solving Quadratics

• As with all equations, solving them simply means
  finding the values that make them true. The
  easiest quadratic equation we could be asked to
  solve would be something like

There are TWO values that make this true x 2
and x -2 (in MATLAB roots(1 0 -4) the value
of the coefficients). We call these the roots of
the equation. In fact this idea of two solutions
extends to just about all quadratic equations.
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Solution by factorization

• At GCSE level you would have spent a lot of time
  practising factorization of quadratic
  expressions. Heres why...
• Consider the following equation
• We can factorise the LHS of the equation as
  follows
• This leaves us with the equation looking like
  this

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Sketching the curve

• Here the solution to renders intercepts on the
  x-axis, while putting x0 gives the intercept on
  the y-axis, hence generating the required three
  points, thus

For a forth point let x1.5 y0.25
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Try one

• Solve the equation hence
  or otherwise sketch the curve

Factorize
x0y-3
x1y-4
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Formula solution to quadratic equations The
process of completing the square leads us to a
formula for solving any quadratic equation. We
can solve a quadratic in the form
With the formula
Those feeling powerful should try to derive this
formula. The starting point should be a general
formula for completing the square.
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The formula for quadratic equations
Examples
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Example solutions
Examples
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The formula for quadratic equations
Note that you MUST rearrange an equation into
ax2 bx c 0 form before using the
formula. Also remember that NOT every quadratic
equation has two solutions, and indeed, some
have none at all!! Well look at the following
examples with the formula.
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Repeated roots
Whats going on here is that the expression x2
6x 9 is a perfect square. Namely (x 3)2. So
we only have one solution to x2 6x 9
0. We say that 3 is a repeated root of the
equation x2 6x 9 0 .
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No Solutions??
Our problem here is that we CANNOT find the
square root of a negative number. In situations
like this, we simply say that the equation has
no solutions. That is, there are NO values of
x that make 3x2 2x 20 equal 0.
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A general approach for solving quadratics
Rearrange into the form Try to factorise. The
idea is that you are so well practised
in factorisation that this shouldnt take too
long! If you cant factorise, use the
formula. Youll note that completing the square
isnt mentioned. Thats why we didnt spend long
talking about it!! The reasons I bothered to
include it at all are (a) the method leads to
the formula, and (b) so that if anyone else
ever mentions it to you, you will know what they
mean, and have somewhere to come back to!
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When is a quadratic not a quadratic?
We will often be faced with equations that dont
look very quadratic, that turn out to be when
rearranged.
All of these can be rearranged-to and/or tackled
as quadratics.
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When its a jar
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Summary
You should now know what is meant by an
equation know what is meant by a quadratic
equation be able to solve a quadratic equation by
factorisation be able to rearrange and solve
without losing a solution be able to use the
completing the square technique know and be able
to use the formula for quadratic equations be
able to recognise (thanks to the formula) when a
quadratic equation has no solutions
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Examination Type Question

• Using suitable math software or otherwise sketch
  the graph of the following functions
• f(x)3x22x-7 6
• g(x)2x3 4
• From the graph obtain approximate values of the
  interception points 2. Use an analytical method
  to obtain better solutions 5. How could you
  obtain better approximations using math software
  3
• 20 marks

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Solution

• Use the MATLAB ginput command to obtain
  approximate solutions

ans -1.7807 -0.6651 1.8386 6.9621
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Analytical solutions

• f(x)3x22x-7
• g(x)2x3

Better solutions could be obtained from MATLAB
ginput command or alternatively as for the roots
of the quadratic
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Further Examination Type Q

• The graph shows two functions namely 4x2-2x1
  x4. Explain which curve is which giving reasons
  for your answer 5 and find the interception
  points 5.

10 marks
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Solution

• The first function is a parabolaB2 (only curveB1)
  due to the polynomial being of second orderB1.
  The second graph is a is a lineB1 as it has the
  form ymxcB1.

M1
M2
A1
A1
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Another Exam Question

• Find the interception of the graphs

and
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M1
M1A1
A1
A1