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PART I : Introduction to Algebra, Systems of
Simultaneous Linear Equations and Quadratic
Equations
1. Introduction to algebra
1.3 Simplification
1. Introduction to algebra
1.1 Representation
Algebra is a system of shorthand → Symbols are used to represent
concepts and variables that are capable of taking different values.
Expression in algebraic form → Not necessary to work out a solution
for every different value of the unknown variable than one is faced
with.
Simplifying an expression means rearranging terms in it so that the
expression becomes easier to work with → General rules for simplification
by type of operations
1.3.1 Addition and Subtraction
Like terms (= same algebraic symbol or symbols, usually multiplied
by a number) can be added or subtracted
All basic arithmetic rules apply when algebraic symbols used
instead of actual numbers
1.2 Evaluation
An expression can be evaluated when variables represented by
algebraic symbols are given specific numerical values.
Jérémie Gross (UNAMUR)
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Consider an expression with only one unknown variable x taking
the following form:
ax 2 + bx + c
(1)
1.3.2 Multiplication
When a set of brackets containing different terms is multiplied by a
symbol or a number, possible to simplify the expression by multiplying
out
Each term in one set of brackets must be multiplied by each term in
the other set.
1.3.3 Factorisation
Transformation of algebraic expression into a format of factors (ie. two
sets of brackets) multiplied together.
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1. Introduction to algebra (Simplification)
1. Introduction to algebra (Simplification)
Jérémie Gross (UNAMUR)
Jérémie Gross (UNAMUR)
Some but not all such expressions can be factorized into sets of
brackets that only involve integers.
No set rules for working out if and how an expression may be
factorized
If the term in x2 does not have a number in front of it (a=1), the
expression can be factorized if there are 2 numbers which:
1
2
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give c when multiplied together and,
give b when added together.
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1. Introduction to algebra (Simplification)
For expressions in the form ax2 + bx + c with a6=1 → Have to find
2 numbers which multiply together give c but also have to find 2
other numbers for coefficients of the 2 terms in x which multiplied
together equal a and allow the coefficient b to be derived when
multiplying out.
1. Introduction to algebra (Simplification)
1.3.4 Division
To divide an algebraic expression by a number → Division of every
terms in the expression by the number, canceling where appropriate.
Specific rules also apply for factorizing expressions with 2 unknown
variables − x and y − of the the form ax2 + bxy + cy2 where a, b
and c are specified parameters → Algebraic Identities can facilitate
factorization of specific types of expressions
Jérémie Gross (UNAMUR)
(x + y )(x − y ) = x 2 − y 2
(2)
(x − y )2 = x 2 − 2xy + y 2
(3)
(x + y )2 = x 2 + 2xy + y 2
(4)
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To divide an expression by another expression (with more than
one term) → Terms can only be cancelled when numerator and
denominator are both multiples of the same factor
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2. Linear Algebra and Solutions of Systems of Simultaneous Equations
1. Introduction to algebra
1.4 Summation sign
2.1 Solving Simple Equations
P
Equation is an algebraic expression that equals a number or another
algebraic expression → Two expressions written on either side of an
equality sign (or an inequality sign)
P
can be used in certain circumstances as a shorthand mean of
expressing the sum of a number of different terms added together:
From 1 to n, the value of index i increases by 1 unit in each successive
term of the expression:
n
X
i = (1) + (2) + ... + (n − 1) + (n)
(5)
i=1
Evaluation of such an expression → Compute the value of each term
separately and then add up.
Jérémie Gross (UNAMUR)
To divide by an unknown variable → Same rule (although it
cannot be simplified any further when the numerator of a fraction
does not contain that variable)
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Elementary principle for solving equations → All terms in the unknown
variable have to be brought together on one side of the equation using the
four balance rules :
1
Equal quantity may be added to both sides of an equation
2
Equal quantity may be subtracted from both sides of an equation
3
Equal quantity may multiply both sides of an equation
4
Equal non-zero quantity may divide both sides of an equation
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2. Linear Algebra and SSSE (Inequality Signs)
2. Linear Algebra and SSSE
When considering inequality relationships, it can be useful to work in
terms of the absolute value of a variable x → written |x| and
defined as
2.2 Inequality Signs
4 inequality signs used in algebra:
>
<
>
6
which
which
which
which
means
means
means
means
’is
’is
’is
’is
always greater than’
always lower than’
greater than or equal to’
lower than or equal to’
1
Attention when using inequality signs with unknown variables
possibly taking negative values (ie. the inequality 2x<3x only holds
if x>0 because if x took a negative value, then the inequality would
be reversed)
2
3
4
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2. Linear Algebra and SSSE
x <0
(6)
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2. Linear algebra
2.4 Systems of Linear Equations
Typical linear (because their graphs are straight lines) equations are
2x1 − 3x2 = 8
General linear system of m equations in n unknowns
(7)
a11 x1 + a12 x2 + ... + a1n xn = b1
a21 x1 + a22 x2 + ... + a2n xn = b2
In general, an equation is linear if it has the form
a1 x1 + a2 x2 + ... + an xn = b
→ Key feature of linear equations is that each term of the equation
contains at most one variable, and that variable is raised only to the
first power rather than to the second, third or some other power.
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...
(8)
Letters a1 , ...,an , and b stand for fixed numbers, such as 2, -3 and 8 in
the second equation and are called parameters. Letters x1 , ..., xn stand
for variables.
Jérémie Gross (UNAMUR)
x >0
when
If both sides multiplied or divided by a positive number, then
direction of the inequality does not change
If both sides multiplied or divided by a negative number, then
direction of the inequality will be reversed.
If both sides squared, the same inequality sign only holds if both
sides are initially positive values (because a negative number
squared becomes a positive number)
If both sides positive and raised to the same negative power, then
direction of the inequality will be reversed.
2.3 Linear Equations
x1 + 2x2 = 3
when
Rules for solving equation with inequality sign → different from
those applied to ordinary equality sign
The last two are sometimes called ’weak inequality’ signs.
Jérémie Gross (UNAMUR)
|x| = x
|x| = −x
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...
...
...
...
(9)
am1 x1 + am2 x2 + ... + amn xn = bm
In which aij and bi are real numbers with aij being the coefficient of
the unknown xj in the ith equation.
2 main questions with respect to a linear system
1
2
A solution does ∃, and if yes
How many solutions are there
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2. Linear Algebra and SSSE (SSLE)
2. Linear Algebra and SSSE
2.5.1 Substitution
2.5 Solutions of Systems of Linear Equations (SSLE)
Elementary ’system of equations’ operations → All reversible:
Adding multiple of one equation to another
Multiplying both sides of an equation by a nonzero scalar
Interchanging two equations
3 main ways of solving systems of linear equations: Substitution
(2.5.1), Elimination of variables (2.5.2) and Matrix methods
(Section 3)
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Method Description Solve one equation of system (9) for one
variable, xn , in terms of the other variables in that equation. Then,
substitute this expression for xn into the other m−1 equations. The
result is a new system of m−1 equations in the n−1 unknowns x1 , ...,
xn−1 . Continue this process by solving one equation in the new
system for xn−1 and substituting this expression into the other m−2
equations to obtain a system of m−2 equations in the n−2 variables
→ Proceed until you reach an easily solvable system with just a single
equation and use the earlier expressions in one variable in terms of the
others to find all the xi ’s.
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2. Linear Algebra and SSSE (SSLE)
2. Linear Algebra and SSSE (SSLE)
2.5.2 Elimination of variables
Example with three-good input-output model with exogenous
demand for 130 units of good 1, 74 units of good 2 and 95 units of
good 3
In/Out X1 X2
X3
X1
0
0.4 0.3
(10)
X2
0.2 0.12 0.14
X3
0.5 0.2 0.05
Entries in the second column declare that it takes 0.4 unit of good 1,
0.12 unit of good 2 and 0.2 unit of good 3 to produce one unit of
good 2
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Method description Use the coefficient of x1 , in the first equation,
to eliminate the x1 terms from all the equations below it. To do this,
add proper multiples of the first equation to each of the succeeding
equations. Then, disregard the first equation and eliminate the next
variable - usually x2 - from the last m−1 equations just as before,
that is, by adding proper multiples of the second equation to each of
the succeeding equations. If the second equation does not contain an
x2 but a lower equation does, you will have to interchange the order
of these two equations before proceeding. Continue eliminating
variables until you reach the last equation. The resulting simplified
system can then easily be solved by substitution.
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2. Linear Algebra and SSSE (SSLE)
2. Linear Algebra and SSSE (SSLE)
Gaussian Elimination Method (GEM) : Method of reducing a given
system of equations by adding a multiple of one equation to another
or by interchanging equations until one reach for the system (9) a
system of the form (11) and then solving (11) via back substitution
Gauss-Jordan Elimination Method (GJEM): Identical to GEM
except that after the system was obtained through GEM, multiply
each equation by a scalar such that 1 be the first non-zero coefficient
of each equation.
x1 − 0.4x2 − 0.3x3 = 130
x2 − 0.25x3 = 125
1x1 − 0.4x2 − 0.3x3 = 130
0.8x2 − 0.2x3 = 100
(12)
x3 = 300
(11)
Apply now eliminating method from the last equation to the first to
eliminate all terms except the term on LHS of each equation.
0.7x3 = 210
Important characteristic of that system is that each equations
contain fewer variables than the previous equations.
x1
= 300
x2
= 200
(13)
x3 = 300
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3. Matrices and Solutions of Systems of Simultaneous Equations
3.1 Introduction to Matrices and SSE
Representation of linear system (9) may be simplified by writing two
rectangular arrays of its coefficients → matrices
Coefficients matrix C of (9)


a11 a12 ... a1n
 a21 a22 ... a2n 

C =
(14)
 ...
... ... ... 
am1 am2 ... amn
Adding on a column corresponding to the RHS in system (9) to
obtain the Augmented matrix A of (9) with rows corresponding
naturally to the equation of (9):


a11 a12 ... a1n b1
 a21 a22 ... a2n b2 

A=
(15)
 ...
... ... ... ... 
am1 am2 ... amn b3
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3. Matrices and SSSE (Introduction to Matrices and SSE)
3 elementary ’equation operations’ become elementary row
operations:
1
2
3
Interchange two rows of a matrix.
Change a row by adding to it a multiple of another row
Multiply each element in a row by the same non-zero number
By applying one or more of those operations to A, the resulting new
augmented matrix, A’, will represent a system of linear equations
which is equivalent (identical solution sets) to the system
represented by the initial augmented matrix A.
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3. Matrices and SSSE (Introduction to Matrices and SSE)
3. Matrices and SSSE (Introduction to Matrices and SSE)
3.1.2 Rank of a Matrix
3.1.1 Row (and Reduced Row) Echelon Forms
DEFINITION: A row of a matrix is said to have leading zeros if the
first k elements of the row are all zeros and the (k + 1)th element of
the row is not zero → Matrix in Row Echelon Form (REF) if each
row has more leading zeros than the row preceding it.
GEM (to get REF) → First nonzero entry in each row of a matrix in
REF called a Pivot
DEFINITION: A row echelon matrix in which each pivot is an identity
(=1) and in which each column containing a pivot contains no other
non-zero entries is said to be in Reduced Row Echelon Form
(RREF).
GJEM (to get RREF) → Row operations to reduce the matrix even
further
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DEFINITION The Rank of a matrix is the number of nonzero rows in
its row echelon form. Note that a row of a matrix is nonzero if and
only if it contains at least one nonzero entry.
3.1.3 Nonsingular Matrix
FACT The Rank of a matrix allows characterizing those coefficient
matrices which have the property that for any RHS b1 , ... , bm , the
corresponding system of linear equations has exactly one solution →
Such coefficient matrices are called nonsingular
DEFINITION A coefficient matrix A is nonsingular, that is, the
corresponding linear system has one and only one solution for every
choice of RHS side b1 , ... , bm if and only if
nber of rows of A = nber of columns of A = rank(A)
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(16)
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3. Matrices and SSSE (Matrix Calculus)
3. Matrices and SSSE
3.2 Matrix Calculus
DEFINITION: A Matrix is simply a rectangular array of numbers.
The size of a matrix is indicated by the number of its rows and the
number of its columns. A matrix with k rows and n columns is called
a k x n (”k” by ”n”) matrix and the number in row i and column j is
called the (i,j)th entry (often written aij ) → Two matrices are equal if
they both have the same size and if the corresponding entries in the
two matrices are equal
Subtraction Since

a11

−  ...
ak1
- A is what one adds to A to obtain 0,
 

. . . a1n
−a11 . . . −a1n
..  =  ..
.. 
aij
.   .
−aij
. 
. . . akn
−ak1 . . . −akn
(17)
Since A - B is just shorthand for A + (- B), we subtract matrices of
the same size simply by subtracting their corresponding entries.
3.2.1 Matrix Algebra
When the sizes (number of rows and columns) are right, 2 matrices can be
added, subtracted, multiplied and even divided.
Jérémie Gross (UNAMUR)
Addition Possible to add 2 matrices of the same size (same number
of row and columns) → Sum will be a new matrix of the same size as
the matrices being added and (i,j)th entry of the sum matrix is simply
the sum of the (i,j)th entries of the 2 matrices being added.
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Scalar multiplication Matrices can be multiplied by ordinary
numbers and the product of the matrix A and the number r, denoted
rA, is the matrix created by multiplying each entry of A by r.
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3. Matrices and SSSE (Matrix Calculus)
3. Matrices and SSSE (Matrix Calculus)
Matrix multiplication Two 6= with algebra of real numbers → Not
all pairs of matrices can be multiplied together and the order in which
matrices are multiplied matters
Matrix product defined if and only if number of columns of A =
number of rows of B → A must be k x m and B must be m x n while
the resulting product matrix AB will be k x n (inherits nber of its row
from A and nber of its columns from B) → To obtain the (i,j)th entry
of AB, multiply the ith row of A and the jth column of B as follows:


b1j


 b2j 
ai1 ai2 ... aim ·  . 
(18)
 .. 
bmj
= ai1 b1j + ai2 b2j + . . . + aim bmj
(19)
For example:


a b
A
 c d ·
C
e f
B
D

aA + bC
=  cA + dC
eA + fC

aB + bD
cB + dD 
eB + fD
The following n x n matrix is called the identity matrix because it is a
multiplicative identity matrix (just as number 1 for real numbers):


1 0 ... 0
 0 1 ... 0 


I = . . .
(21)

. . ... 
 .. ..
0 0 ... 1
with aii =1 for all i and aij =0 for all i 6= j has the property that for any
m x n matrix A
AI = A
(22)
and for any n x l matrix B
IB = B
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(20)
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(23)
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3. Matrices and SSSE (Matrix Calculus)
3. Matrices and SSSE (Matrix Calculus)
3.2.2 Matrix Transpostion
Laws of matrix algebra
Associative laws:
(A + B) + C = A + (B + C )
(24)
(AB)C = A(BC )
(25)
The Transpose of a k x n matrix A is the n x k matrix AT obtained
by interchanging the rows and columns of A → The (i,j)th entry of A
becomes the (j,i)th entry of AT
Rules for transposition:
Commutative law for addition:
A+B =B +A
(26)
Distributive laws:
A(B + C ) = AB + AC
(27)
(A + B)C = AC + BC
(28)
(A + B)T = AT + B T
(29)
(A − B)T = AT − B T
(30)
(AT )T = A
(31)
(rA)T = rAT
(32)
where A and B are k x n matrices and r is a scalar.
(AB)T = B T AT
(33)
where A be a k x m matrix and B be an m x n matrix.
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3. Matrices and SSSE (Matrix Calculus)
3.2.3 The determinant
3. Matrices and SSSE (Matrix Calculus)
Determinant for 1 x 1 matrices (= scalar)
det(a) = a
Determinant for 2 x 2 matrices
a11 a12
A=
→ det(A) = a11 a22 − a12 a21
a21 a22
(34)
(35)
To motivate the general definition of a determinant, write det(A) as:
det(A) = a11 det(a22 ) − a12 det(a21 )
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is called the (i,j)th Minor of A and the scalar
Cij = (−1)i+j Mij
(36)
1st term on the RHS is the (1,1)th entry of A times the det of the
submatrix obtained by deleting from A the row and the column which
contain that entry. 2nd term is the (1,2)th entry times the det of the
submatrix obtained by deleting from A the row and the column which
contain that entry. Terms alternate in signs: the term containing a11
receives + sign and the term containing a12 receives - sign.
Jérémie Gross (UNAMUR)
DEFINITION Let A be an n x n matrix. Let Aij be the (n−1) x
(n−1) submatrix obtained by deleting row i and column j from A.
Then, the scalar
Mij = det(Aij )
(37)
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is called the
the scalar
(i,j)th
Jérémie Gross (UNAMUR)
(38)
Cofactor (corresponds to signed minor) of A and
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3. Matrices and SSSE (Matrix Calculus)
Determinant for

a11
det  a21
a31
3 x 3 matrices

a12 a13
a22 a23  = a11 M11 − a12 M12 + a13 M13
a32 a33
3. Matrices and SSSE (Matrix Calculus)
(39)
= a11 C11 + a12 C12 + a13 C13
(40)
a21 a23
a21 a22
= a11 ·det
−a12 ·det
+a13 ·det
a31 a33
a31 a32
(41)
Determinant for n x n matrices A
PROPERTY A square matrix is nonsingular if and only if its
determinant is nonzero
Algebraic Properties of Determinants:
a22 a23
a32 a33
det(A) = a11 C11 + a12 C12 + . . . + a1n C1n
det(AT ) = det(A)
(43)
det(A · B) = det(A) · det(B)
(44)
det(A + B) 6= det(A) + det(B)
(45)
(42)
Nothing special about the first row → Use any row or column to
compute the determinant of a matrix
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3. Matrices and SSSE (Matrix Calculus)
3.2.4 Matrix inversion
DEFINITION For any n x n matrix A, let Cij denote the (i,j)th
cofactor of A, that is (-1)i+j times the determinant of the submatrix
obtained by deleting row i and column j from A. The n x n matrix
whose (i,j)th entry is Cji - the (j,i)th cofactor of A - is called the
Adjoint of A and is written adj(A).
THEOREM Let A be a nonsingular matrix (det(A)6=0), then its
inverse A−1 is
1
A−1 =
· adj(A)
(46)
det(A)
Example of using theorem to invert

a
A= d
g
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3. Matrices and SSSE (Matrix Calculus)
THEOREM Let A and B be square invertible matrices. Then,
(A−1 )−1 = A
(48)
T −1
(49)
(A )
= (A−1 )T
AB is invertible and (AB)−1 = B −1 A−1
(50)
matrix

b c
e f 
h i
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(47)
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3. Matrices and SSSE
3. Matrices and SSSE (Matrix Methods to solve for SSE)
3.3 Matrix Methods to solve for SSE
Consider a system of 2 equations in 2 variables x and y:
ax1 + bx2 = u
(51)
cx1 + dx2 = v
(52)
Two ways to solve for x and y using matrix calculus → Cramer’s rule
(3.3.1) and Method using Matrix Inversion (3.3.2)
3.3.1 Cramer’s Rule
RULE: Unique solution x=(x1 ,..., xn ) of the n x n system Ax=b is :
det(Bi )
xi =
for i = 1, ..., n
det(A)
ILLUSTRATION Write the 2 equations of the system in matrix form
as
a b
x1
u
=
(54)
c d
x2
v
Cramer’s rule says that the solutions are given by
u b
det
v d
ud − bv
=
x1 =
det(A)
ad − bc
and
det
(53)
x2 =
a u
c v
det(A)
(55)
=
va − cu
ad − bc
(56)
where Bi is matrix A with the RHS B replacing the ith column of A.
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3. Matrices and SSSE (Matrix Methods to solve for SSE)
4. Quadratic Equations
3.3.2 Method using Matrix Inversion
4.1 Quadratic Equations: Characterization
STATEMENT If an n x n matrix A is invertible, then it is nonsingular,
and the unique solution to the system of linear equations Ax=b is
x = A−1 b
(57)
ILLUSTRATION Write the 2 equations of the system in matrix form
(as when using Cramer’s rule) and solve by inverting the matrix on
the LHS → The inverse of this matrix is
1
d −b
(58)
−c a
ad − bc
and allows to obtain the following expression
1
x1
d −b
u
=
x2
−c a
v
ad − bc
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ax 2 + bx + c = 0
(60)
where x is an unknown variable and a, b, and c are constant
parameters with a6=0
→ Every quadratic equation that can be solved has 2 solutions called
Roots
4.2 Methods to solve for Quadratic Equations
(59)
from which the unique solution of the system is easily derived.
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A Quadratic Equation includes necessarily a quadratic expression
(variable that is squared) and takes the form
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3 possible methods for solving unknown x in a quadratic equation: By
factorization (4.2.1), by using the quadratic ’formula’ (4.2.2) and by
plotting a graph (PART II)
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AMIDE - 2016/2017 - C&M - QM - PI
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4. Quadratic Equations (Methods to solve for Quadratic Equations)
3. Quadratic equations (Methods to solve for Quadratic Equations)
4.2.1 Factorization
4.2.2 Quadratic Formula
Factorization allows to break down some expressions into terms
which when multiplied together give the original expression. For
example,
a2 − 2ab + b 2 = (a − b)(a − b)
(61)
If a quadratic function rearranged to equal zero can be factorized in
this way → One or the other of the two factors must equal zero
(A·B=0 then either A, B or both must be zero)
Only useful as a short-cut way of solving certain quadratic
equations → If you cannot quickly see a way of factorizing then you
should use the formula method
Jérémie Gross (UNAMUR)
AMIDE - 2016/2017 - C&M - QM - PI
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Any Quadratic Equation expressed in the form
ax 2 + bx + c = 0
(62)
where a, b, and c are given parameters and for which a solution exists can
be solved for x by using the Quadratic Formula
√
−b ± b 2 − 4ac
x1,2 =
(63)
2a
Jérémie Gross (UNAMUR)
AMIDE - 2016/2017 - C&M - QM - PI
40 / 41
4. Quadratic Equations
4.3 Systems of Simultaneous Quadratic Equations
If one or more equations in a simultaneous equation system are quadratic
then it may be possible to eliminate all but one unknown and to reduce
the problem to a Single Quadratic Equation → If this can be solved
then the other unknowns can be found by substitution
Jérémie Gross (UNAMUR)
AMIDE - 2016/2017 - C&M - QM - PI
41 / 41