4 eso academics - unit 04 - equations and inequations
TRANSCRIPT
Unit 04 December
1. EQUATIONS.
An Equation is a statement using algebra that contains an unknown quantity
and an equals sign. The Solution of an equation is the set of values which, when
substituted for unknowns, make the equation a true statement. An Equation has
different Elements:
β’ Variables: The unknown quantities
β’ Member: The two expressions on either side of an equation.
β’ Term: Any of the addends of an equation.
β’ Degree: For a term with one variable, the degree is the variable's exponent. With
more than one variable, the degree is the sum of the exponents of the variables.
MATH VOCABULARY: Equation, Member, Term, Variable, Coefficient, Degree,
Constant, Solution.
Axel CotΓ³n GutiΓ©rrez Mathematics 4ΒΊ ESO 4.4.1
Unit 04 December
2. FIRST-DEGREE AND SECOND DEGREE EQUATIONS.
2.1. FIRST-DEGREE EQUATIONS.
A First-Degree Equation is called a Linear Equation. The highest exponent of a
linear equation is 1. The standard form for a linear equation is:
ππ,ππ, ππ β β€ ππππππ ππ β ππ
3π₯π₯ + 2 = β10.
The solution of this equation is π₯π₯ = β4
2.2. SECOND-DEGREE EQUATIONS.
A Second-Degree Equation is called a Quadratic Equation. The highest
exponent of a quadratic equation is 2. The standard form for a quadratic equation is:
ππ,ππ, ππ β β€ ππππππ ππ β ππ
5π₯π₯2 + 5π₯π₯ β 2 = 0
Quadratic equations can be solved using a special formula called the Quadratic
Formula:
The solutions to the quadratic equation are often called Roots, or sometimes
Zeroes.
Axel CotΓ³n GutiΓ©rrez Mathematics 4ΒΊ ESO 4.4.2
Unit 04 December
Case 1:
π₯π₯2 β 6π₯π₯ + 5 = 0 βΉ ππ = 1; ππ = β6 ππππππ ππ = 5
π₯π₯ =βππ Β± βππ2 β 4ππππ
2ππ=β(β6) Β± οΏ½(β6)2 β (4 β 1 β 5)
2 β 1=
=6 Β± β36 β 20
2=
6 Β± β162
=6 Β± 4
2=
3 Β± 21
= 3 Β± 2 = οΏ½3 + 2 = 53 β 2 = 1
This equation has two different solutions: π₯π₯1 = 5 ππππππ π₯π₯2 = 1
Case 2:
4π₯π₯2 + 4π₯π₯ + 1 = 0 βΉ ππ = 4; ππ = 4 ππππππ ππ = 1
π₯π₯ =βππ Β± βππ2 β 4ππππ
2ππ=β(+4) Β± οΏ½(+4)2 β (4 β 4 β 1)
2 β 4=
=β4 Β± β16 β 16
8=β4 Β± β0
8=β4 Β± 0
8= οΏ½
β48
= β12
β48
= β12
Both solutions are equal. We say that this equation has a double solution or double
root.
Case 3:
π₯π₯2 β 2π₯π₯ + 5 = 0 βΉ ππ = 1; ππ = β2 ππππππ ππ = 5
π₯π₯ =βππ Β± βππ2 β 4ππππ
2ππ=β(β2) Β± οΏ½(β2)2 β (4 β 1 β 5)
2 β 1=
=2 Β± β4 β 20
2=
2 Β± ββ162
=
β©βͺβ¨
βͺβ§2 + ββ16
2= βπ₯π₯ β β
2 β ββ162
= βπ₯π₯ β β
This equation has no solution into the set of real numbers.
Axel CotΓ³n GutiΓ©rrez Mathematics 4ΒΊ ESO 4.4.3
Unit 04 December
The previous examples show that the different types of solutions of the second-
degree equations depend on the value of π«π« = ππππ β ππππππ. This number is called the
Discriminant.
Sometimes you can find βIncompleteβ quadratic equations, if ππ = ππ π¨π¨π¨π¨ ππ =
ππ; π¨π¨π¨π¨ ππ = ππ ππππππ ππ = ππ. You can solve these equations in an easy way, without using the
quadratic formula.
Axel CotΓ³n GutiΓ©rrez Mathematics 4ΒΊ ESO 4.4.4
Unit 04 December
MATH VOCABULARY: First-Degree Equation, Linear Equation, Second-Degree Equation,
Quadratic Equation, Quadratic Formula, Root, Zero, Double Root, Discriminant,
Incomplete Quadratic Equation.
3. OTHER TYPES OF EQUATIONS.
3.1. BIQUADRATIC EQUATIONS.
Biquadratic Equations are quartic equations with no odd-degree terms. The
standard form for a biquadratic equation is:
LetΒ΄s see the case of fourth degree:
These equations can be written as a quadratic equation by making an
appropriate substitution called Change of variable:
To solve we have only to apply the Quadratic Formula:
Axel CotΓ³n GutiΓ©rrez Mathematics 4ΒΊ ESO 4.4.5
Unit 04 December
The four roots of the equation will be:
Solve: π₯π₯4 β 3π₯π₯2 β 4 = 0
π₯π₯2 = π‘π‘
π‘π‘2 β 3π‘π‘ β 4 = 0
We apply quadratic formula:
π‘π‘ =β(β3) Β± οΏ½(β3)2 β οΏ½4 β 1 β (β4)οΏ½
2 β 1=
3 Β± β252
= οΏ½ π‘π‘1 = 4π‘π‘2 = β1
β
β©βͺβ¨
βͺβ§ π₯π₯1 = +β4 = 2
π₯π₯2 = ββ4 = β2π₯π₯3 = +ββ1 = βπ₯π₯ β βπ₯π₯4 = βββ1 = βπ₯π₯ β β
MATH VOCABULARY: Biquadratic equations, Change of Variable.
3.2. EQUATIONS WHICH CAN BE SOLVED BY FACTORIZING.
We can solve an algebraic expression by factorizing if we can write it as the
product of factors.
Axel CotΓ³n GutiΓ©rrez Mathematics 4ΒΊ ESO 4.4.6
Unit 04 December
π·π·ππ(ππ) β π·π·ππ(ππ) β β¦ β π·π·ππ(ππ) = ππ
We can do it extracting factors or with RuffiniΒ΄s rule as the unit 3.
Solve π₯π₯4 β 5π₯π₯2 = π₯π₯3 β π₯π₯2 β 4x
Firstly we relocate the equation:
π₯π₯4 β π₯π₯3 β 4π₯π₯2 + 4x = 0
We extract x as a common factor:
π₯π₯(π₯π₯3 β π₯π₯2 β 4π₯π₯ + 4) = 0
Now we use RuffiniΒ΄s rule:
1 β1 β4 +4
1 +1 0 β4
1 0 β4 0
β π₯π₯(π₯π₯3 β π₯π₯2 β 4π₯π₯ + 4) = π₯π₯(π₯π₯ β 1)(π₯π₯2 β 4)
By using polynomial identities:
π₯π₯4 β 5π₯π₯2 = π₯π₯3 β π₯π₯2 β 4x = π₯π₯(π₯π₯ β 1)(π₯π₯ + 2)(π₯π₯ β 2)
The solutions are:
β οΏ½
π₯π₯1 = 0π₯π₯2 = 1π₯π₯3 = β2π₯π₯4 = 2
Axel CotΓ³n GutiΓ©rrez Mathematics 4ΒΊ ESO 4.4.7
Unit 04 December
3.3. RATIONAL EQUATIONS.
Equations that contain rational expressions are called Rational Equations. We
solve rational equations by finding a common denominator. We can follow either of
two methods:
β’ We can rewrite the equation so that all terms have the common denominator and
we can solve for the variable with just the numerators.
β’ Or we can multiply both sides of the equation by the common denominator so
that all terms become polynomials instead of rational expressions.
An important step in solving rational equations is to reject any Extraneous
Solutions from the final answer. Extraneous solutions are solutions that donβt satisfy
the original form of the equation because they produce untrue statements or are
excluded values that make a denominator equal to 0. Therefore, we should always
check solutions in the original equation.
Solve:
π₯π₯π₯π₯ β 3
+3
π₯π₯ + 1= β1
πΏπΏπΆπΆπΆπΆ(π₯π₯ β 3, π₯π₯ + 1) = (π₯π₯ β 3)(π₯π₯ + 1)
π₯π₯(π₯π₯ + 1)(π₯π₯ β 3)(π₯π₯ + 1) +
3(π₯π₯ β 3)(π₯π₯ β 3)(π₯π₯ + 1) =
β1(π₯π₯ β 3)(π₯π₯ + 1)(π₯π₯ β 3)(π₯π₯ + 1)
π₯π₯(π₯π₯ + 1) + 3(π₯π₯ β 3) = β1(π₯π₯ β 3)(π₯π₯ + 1)
π₯π₯2 + π₯π₯ + 3π₯π₯ β 9 = βπ₯π₯2 β π₯π₯ + 3π₯π₯ + 3
2π₯π₯2 + 2π₯π₯ β 12 = π₯π₯2 + π₯π₯ β 6 = 0
Axel CotΓ³n GutiΓ©rrez Mathematics 4ΒΊ ESO 4.4.8
Unit 04 December
π₯π₯ =β1 Β± β1 + 24
2= οΏ½ π₯π₯1 = 2
π₯π₯2 = β3
We have to check the answers:
π₯π₯π₯π₯ β 3
+3
π₯π₯ + 1= β1
π₯π₯=2οΏ½οΏ½οΏ½
22 β 3
+3
2 + 1=
2β1
+33
= β1 βΉ πππ‘π‘Β΄π π ππ π π πππππππ‘π‘ππππππ
π₯π₯π₯π₯ β 3
+3
π₯π₯ + 1= β1
π₯π₯=β3οΏ½οΏ½οΏ½οΏ½
β3β3 β 3
+3
β3 + 1=β3β6
+3β2
= β1 βΉ πππ‘π‘Β΄π π ππ π π πππππππ‘π‘ππππππ
MATH VOCABULARY: Rational equations, Extraneous Solutions.
3.4. RADICAL EQUATIONS.
A Radical Equation is an equation in which the variable is contained inside a
radical symbol (in the radicand). To solve it we:
β’ Isolate the radical on one side of the equation.
β’ If the radical is a square root, square each side of the equation. (If the radical is
not a square root, raise each side to a power equal to the index of the root).
β’ Simplify and solve as you would any equations.
β’ Substitute answers back into the original equation to make sure that your
solutions are valid (there could be some extraneous roots that do not satisfy the
original equation and that you must throw out).
Solve: βπ₯π₯ + 5 β β8 β π₯π₯ = β1
βπ₯π₯ + 5 = β1 + β8 β π₯π₯
Squaring both members:
Axel CotΓ³n GutiΓ©rrez Mathematics 4ΒΊ ESO 4.4.9
Unit 04 December
οΏ½βπ₯π₯ + 5οΏ½2
= οΏ½β1 + β8 β π₯π₯οΏ½2
π₯π₯ + 5 = 1 β 2β8 β π₯π₯ + 8 β π₯π₯
2π₯π₯ β 4 = β2β8 β π₯π₯
π₯π₯ β 2 = ββ8 β π₯π₯
2 β π₯π₯ = β8 β π₯π₯
Squaring again:
(2 β π₯π₯)2 = οΏ½β8 β π₯π₯οΏ½2
4 β 4π₯π₯ + π₯π₯2 = 8 β π₯π₯
π₯π₯2 β 3π₯π₯ β 4 = 0
π₯π₯ =3 Β± β9 + 16
2= οΏ½ π₯π₯1 = 4
π₯π₯2 = β1
We have to check the answers:
βπ₯π₯ + 5 β β8 β π₯π₯ = β1π₯π₯=4οΏ½οΏ½οΏ½β4 + 5 β β8 β 4 = 3 β 2 = 1 β β1
βΉ πΌπΌπ‘π‘ πππ π πππππ‘π‘ ππ π π πππππππ‘π‘ππππππ
βπ₯π₯ + 5 β β8 β π₯π₯ = β1π₯π₯=β1οΏ½οΏ½οΏ½οΏ½ββ1 + 5 βοΏ½8 β (β1) = 2 β 3 = β1
βΉ πππ‘π‘Β΄π π ππ π π πππππππ‘π‘ππππππ
MATH VOCABULARY: Radical equations, To Isolate.
Axel CotΓ³n GutiΓ©rrez Mathematics 4ΒΊ ESO 4.4.10
Unit 04 December
4. INEQUATIONS.
An Inequation, also known as an Inequality, is an algebraic expression
containing an inequality sign (<, >,β€,β₯).
2π₯π₯ β 4 > β2; π₯π₯2 + 5 β€ 0
The solution to an inequation is the set of values that make true the inequality.
We will see two methods for solving inequations: graphical method and algebraic
method.
4.1. GRAPHICAL METHOD.
We have to draw the graph of the inequation as if they were equations and
then set the inequalities.
Case 1:
2x + 4 > 0
It you draw the graph of the line π¦π¦ = 2π₯π₯ + 4, the solution to the inequation will be the
set of values of π₯π₯ such that π¦π¦ > 0
Axel CotΓ³n GutiΓ©rrez Mathematics 4ΒΊ ESO 4.4.11
Unit 04 December
Looking at the graph, you can see that when π₯π₯ > β2, then π¦π¦ > 0.
That is, the solution to the inequation 2x + 4 > 0 is the interval (β2,β).
Case 2:
β2x + 7 >π₯π₯2β 3
The ordinate of the line π¦π¦ = β2x + 7 is greater than or equal to the ordinate of the
line π¦π¦ = π₯π₯2β 3 for the values of π₯π₯ less than or equal to 4.
The solution to the inequation is the interval (ββ, 4)
Case 3:
βπ₯π₯2 + 4π₯π₯ > 2π₯π₯ β 3
The ordinate of the parabola π¦π¦ = βπ₯π₯2 + 4π₯π₯ is greater than or equal to the ordinate of
the line π¦π¦ = 2π₯π₯ β 3 for the values of π₯π₯ between β1 and 3.
The solution to the inequation is the interval (β3,4)
Axel CotΓ³n GutiΓ©rrez Mathematics 4ΒΊ ESO 4.4.12
Unit 04 December
In general, to solve an inequation with one variable, ππ(ππ) β€ ππ(ππ) or
ππ(ππ) β€ ππ(ππ) :
β’ Represent the graphs of the functions ππ = ππ(ππ) and ππ = ππ(ππ) , and get the points
where they intersect.
β’ Find the intervals that make true the inequality.
4.2. ALGEBRAIC METHOD.
Solving inequations is similar to solving equations. We add, subtract, multiply or
divide both sides until we get the variable by itself. There is one important difference
though. With inequations we can add or subtract the same thing to both sides and the
inequality stays the same. We can multiply or divide both sides by a positive number
and the inequality stays the same. But if we multiply by a negative number, then the
sign of the inequality Reverses.
Solve:
Axel CotΓ³n GutiΓ©rrez Mathematics 4ΒΊ ESO 4.4.13
Unit 04 December
β2x β 7 <π₯π₯2
+ 3
β4π₯π₯ β 14 < π₯π₯ + 6
β4π₯π₯ β π₯π₯ < 6 + 14
β5π₯π₯ < 20 β π₯π₯ >20β5
= β4
The solution is the interval (β4,β)
Solve:
π₯π₯2 β 5π₯π₯ β 6 β€ βπ₯π₯ β 1
π₯π₯2 β 4π₯π₯ β 5 β€ 0
We find the roots of the polynomial π₯π₯2 β 4π₯π₯ β 5
π₯π₯1 = β1 ππππππ π₯π₯2 = 5
We plot these roots on the number line and find what is the sign of the number value
of the polynomial π₯π₯2 β 4π₯π₯ β 5 in each of the intervals in which this line has been split.
We take any value of these intervals, for instance:
If πΌπΌπΌπΌ π₯π₯ = β2 β (β2)2 β 4(β2) β 5 = 7 > 0
Axel CotΓ³n GutiΓ©rrez Mathematics 4ΒΊ ESO 4.4.14
Unit 04 December
If πΌπΌπΌπΌ π₯π₯ = 0 β (0)2 β 4(0) β 5 = 7 β 5 < 0
If πΌπΌπΌπΌ π₯π₯ = 6 β (6)2 β 4(6) β 5 = 7 > 0
The solution to the inequation is the interval [β1,5]
Axel CotΓ³n GutiΓ©rrez Mathematics 4ΒΊ ESO 4.4.15