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: • Variables: The unknown quantities • Member: The two expressions on either side of an equation. • Term: Any of the addends of an equation. • 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 Cotón Gutiérrez Mathematics 4º ESO 4.4.1
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4 ESO Academics - UNIT 04 - EQUATIONS AND INEQUATIONS
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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:
• Variables: The unknown quantities
• Member: The two expressions on either side of an equation.
• Term: Any of the addends of an equation.
• 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 Cotón Gutiérrez Mathematics 4º 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:
𝒂𝒂,𝒃𝒃, 𝒄𝒄 ∈ ℤ 𝒂𝒂𝒂𝒂𝒂𝒂 𝒂𝒂 ≠ 𝟎𝟎
3𝑥𝑥 + 2 = −10.
The solution of this equation is 𝑥𝑥 = −4
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:
𝒂𝒂,𝒃𝒃, 𝒄𝒄 ∈ ℤ 𝒂𝒂𝒂𝒂𝒂𝒂 𝒂𝒂 ≠ 𝟎𝟎
5𝑥𝑥2 + 5𝑥𝑥 − 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 Cotón Gutiérrez Mathematics 4º ESO 4.4.2
Unit 04 December
Case 1:
𝑥𝑥2 − 6𝑥𝑥 + 5 = 0 ⟹ 𝑎𝑎 = 1; 𝑏𝑏 = −6 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐 = 5
𝑥𝑥 =−𝑏𝑏 ± √𝑏𝑏2 − 4𝑎𝑎𝑐𝑐
2𝑎𝑎=−(−6) ± �(−6)2 − (4 ∙ 1 ∙ 5)
2 ∙ 1=
=6 ± √36 − 20
2=
6 ± √162
=6 ± 4
2=
3 ± 21
= 3 ± 2 = �3 + 2 = 53 − 2 = 1
This equation has two different solutions: 𝑥𝑥1 = 5 𝑎𝑎𝑎𝑎𝑎𝑎 𝑥𝑥2 = 1
Case 2:
4𝑥𝑥2 + 4𝑥𝑥 + 1 = 0 ⟹ 𝑎𝑎 = 4; 𝑏𝑏 = 4 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐 = 1
𝑥𝑥 =−𝑏𝑏 ± √𝑏𝑏2 − 4𝑎𝑎𝑐𝑐
2𝑎𝑎=−(+4) ± �(+4)2 − (4 ∙ 4 ∙ 1)
2 ∙ 4=
=−4 ± √16 − 16
8=−4 ± √0
8=−4 ± 0
8= �
−48
= −12
−48
= −12
Both solutions are equal. We say that this equation has a double solution or double
root.
Case 3:
𝑥𝑥2 − 2𝑥𝑥 + 5 = 0 ⟹ 𝑎𝑎 = 1; 𝑏𝑏 = −2 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐 = 5
𝑥𝑥 =−𝑏𝑏 ± √𝑏𝑏2 − 4𝑎𝑎𝑐𝑐
2𝑎𝑎=−(−2) ± �(−2)2 − (4 ∙ 1 ∙ 5)
2 ∙ 1=
=2 ± √4 − 20
2=
2 ± √−162
=
⎩⎪⎨
⎪⎧2 + √−16
2= ∄𝑥𝑥 ∈ ℝ
2 − √−162
= ∄𝑥𝑥 ∈ ℝ
This equation has no solution into the set of real numbers.
Axel Cotón Gutiérrez Mathematics 4º 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 𝑫𝑫 = 𝒃𝒃𝟐𝟐 − 𝟒𝟒𝒂𝒂𝒄𝒄. This number is called the
Discriminant.
Sometimes you can find “Incomplete” quadratic equations, if 𝐛𝐛 = 𝟎𝟎 𝐨𝐨𝐨𝐨 𝐜𝐜 =
𝟎𝟎; 𝐨𝐨𝐨𝐨 𝐚𝐚 = 𝟎𝟎 𝐚𝐚𝐚𝐚𝐚𝐚 𝐛𝐛 = 𝟎𝟎. You can solve these equations in an easy way, without using the
quadratic formula.
Axel Cotón Gutiérrez Mathematics 4º ESO 4.4.4
Unit 04 December
MATH VOCABULARY: First-Degree Equation, Linear Equation, Second-Degree Equation,