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This comprehensive guide covers essential algebraic formulas and geometric formulas for students, including special products, quadratic equations, means, progressions, and various geometric shapes. It provides a thorough overview of key concepts in basic algebra and geometry with clear explanations and formulas.
8/11/2023
390
This page presents volume and surface area formulas for various three-dimensional shapes, essential for students studying solid geometry.
The following shapes are covered:
Definition: A frustum is the portion of a solid (usually a cone or pyramid) lying between two parallel planes cutting the solid.
The page also includes formulas for spheres and related shapes:
Example: For a sphere with radius 5 cm:
This page introduces fundamental algebraic formulas for special products and quadratic equations, essential for students studying basic algebra.
The special product formulas cover various algebraic expansions, including:
The quadratic formula for solving equations in the form ax² + bx + c = 0 is presented: x = [-b ± √(b² - 4ac)] / (2a)
Definition: The discriminant (D = b² - 4ac) determines the nature of the roots:
Highlight: Understanding these special product formulas and the quadratic formula is crucial for solving a wide range of algebraic problems in higher mathematics.
This page focuses on area formulas for various types of quadrilaterals, expanding on geometric formulas for plane figures.
The following quadrilaterals are covered:
Vocabulary: A kite is a quadrilateral with two pairs of adjacent sides that are equal in length.
For general quadrilaterals, more advanced formulas are presented:
Highlight: Ptolemy's theorem for cyclic quadrilaterals states that the product of the diagonals equals the sum of the products of opposite sides: d₁d₂ = ac + bd.
The page concludes with formulas for quadrilaterals circumscribing a circle and the area given diagonals and their angle of intersection.
This page presents formulas related to circles and regular polygons, essential components of basic geometry.
For circles, the following formulas are provided:
Definition: A sector of a circle is the region bounded by two radii and an arc of the circle.
For regular polygons, the page includes formulas for:
Example: For a regular hexagon (n = 6):
The page also provides relationships between the apothem, side length, and circumradius of a regular polygon.
This page focuses on various formulas for calculating the area of triangles, a fundamental topic in geometry.
The following area formulas are presented:
Highlight: Heron's formula is particularly useful when only the side lengths of a triangle are known, while the SAS formula is helpful when an angle and two sides are given.
The page also includes a formula for calculating the length of a median in a triangle and the area of a five-pointed star.
Vocabulary: The median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side.
This page covers Vieta's formulas for polynomial equations and introduces different types of means, expanding on basic algebra and geometry formulas.
Vieta's formulas are presented for quadratic, cubic, and quartic equations, relating the coefficients of a polynomial to sums and products of its roots.
Definition: Vieta's formulas provide a way to express the sums and products of the roots of a polynomial equation in terms of its coefficients.
The page then introduces four types of means:
Example: For the numbers 2, 4, and 8:
The page concludes with formulas for arithmetic and geometric progressions, including terms and sums of finite and infinite series.
This page delves deeper into algebraic formulas, focusing on binomial expansions and clock problems.
The binomial expansion formula is presented for (ax + by)ⁿ, including:
For clock problems, a formula is provided to calculate the angle between the hour and minute hands:
Angle = |30H - 5.5M|, where H is the hour and M is the minutes
Example: In a clock problem, to find the time when the hands form a right angle (90°), you would solve the equation: |30H - 5.5M| = 90°
The page also introduces the multinomial expansion formula for expressions like (Ax + By + Cz)ⁿ.
Vocabulary: Multinomial expansion extends the concept of binomial expansion to expressions with more than two terms.
This page continues with analytic geometry, covering triangles defined by coordinates and introducing conic sections.
For triangles defined by coordinates (x₁, y₁), (x₂, y₂), and (x₃, y₃), the following are presented:
Vocabulary: The centroid of a triangle is the point where its three medians intersect, dividing each median in a 2:1 ratio.
The page then introduces the general form of conic sections: Ax² + Bxy + Cy² + Dx + Ey + F = 0
When B = 0, the equation simplifies to: Ax² + Cy² + Dx + Ey + F = 0
Definition: Conic sections are curves formed by the intersection of a plane and a double cone. They include circles, ellipses, parabolas, and hyperbolas.
The type of conic section can be determined based on the coefficients:
Example: The equation x² + y² = 25 represents a circle with radius 5 centered at the origin.
Highlight: Understanding conic sections is crucial for many applications in physics, engineering, and advanced mathematics.
This page provides information on the properties of regular polyhedra, also known as Platonic solids, which are important in solid geometry.
The five Platonic solids are presented with their properties:
For each polyhedron, the following properties are listed:
Highlight: Euler's formula for polyhedra that do not intersect themselves: F + V - E = 2, where F is the number of faces, V is the number of vertices, and E is the number of edges.
Example: For a cube with side length s:
Vocabulary: A regular polyhedron is a three-dimensional solid where all faces are congruent regular polygons, and all vertices are surrounded by the same number of faces.
This page introduces fundamental concepts of analytic geometry, focusing on equations of lines and distance formulas.
Various forms of line equations are presented:
Definition: The slope (m) of a line is the change in y divided by the change in x between any two points on the line.
Distance formulas are provided for:
The page also includes formulas for parallel lines and the angle of inclination of a line.
Example: The distance between points (1, 2) and (4, 6) is: d = √[(4 - 1)² + (6 - 2)²] = √(3² + 4²) = 5
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SuSSan, iOS User
This comprehensive guide covers essential algebraic formulas and geometric formulas for students, including special products, quadratic equations, means, progressions, and various geometric shapes. It provides a thorough overview of key concepts in basic algebra and geometry with clear explanations and formulas.
This page presents volume and surface area formulas for various three-dimensional shapes, essential for students studying solid geometry.
The following shapes are covered:
Definition: A frustum is the portion of a solid (usually a cone or pyramid) lying between two parallel planes cutting the solid.
The page also includes formulas for spheres and related shapes:
Example: For a sphere with radius 5 cm:
This page introduces fundamental algebraic formulas for special products and quadratic equations, essential for students studying basic algebra.
The special product formulas cover various algebraic expansions, including:
The quadratic formula for solving equations in the form ax² + bx + c = 0 is presented: x = [-b ± √(b² - 4ac)] / (2a)
Definition: The discriminant (D = b² - 4ac) determines the nature of the roots:
Highlight: Understanding these special product formulas and the quadratic formula is crucial for solving a wide range of algebraic problems in higher mathematics.
This page focuses on area formulas for various types of quadrilaterals, expanding on geometric formulas for plane figures.
The following quadrilaterals are covered:
Vocabulary: A kite is a quadrilateral with two pairs of adjacent sides that are equal in length.
For general quadrilaterals, more advanced formulas are presented:
Highlight: Ptolemy's theorem for cyclic quadrilaterals states that the product of the diagonals equals the sum of the products of opposite sides: d₁d₂ = ac + bd.
The page concludes with formulas for quadrilaterals circumscribing a circle and the area given diagonals and their angle of intersection.
This page presents formulas related to circles and regular polygons, essential components of basic geometry.
For circles, the following formulas are provided:
Definition: A sector of a circle is the region bounded by two radii and an arc of the circle.
For regular polygons, the page includes formulas for:
Example: For a regular hexagon (n = 6):
The page also provides relationships between the apothem, side length, and circumradius of a regular polygon.
This page focuses on various formulas for calculating the area of triangles, a fundamental topic in geometry.
The following area formulas are presented:
Highlight: Heron's formula is particularly useful when only the side lengths of a triangle are known, while the SAS formula is helpful when an angle and two sides are given.
The page also includes a formula for calculating the length of a median in a triangle and the area of a five-pointed star.
Vocabulary: The median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side.
This page covers Vieta's formulas for polynomial equations and introduces different types of means, expanding on basic algebra and geometry formulas.
Vieta's formulas are presented for quadratic, cubic, and quartic equations, relating the coefficients of a polynomial to sums and products of its roots.
Definition: Vieta's formulas provide a way to express the sums and products of the roots of a polynomial equation in terms of its coefficients.
The page then introduces four types of means:
Example: For the numbers 2, 4, and 8:
The page concludes with formulas for arithmetic and geometric progressions, including terms and sums of finite and infinite series.
This page delves deeper into algebraic formulas, focusing on binomial expansions and clock problems.
The binomial expansion formula is presented for (ax + by)ⁿ, including:
For clock problems, a formula is provided to calculate the angle between the hour and minute hands:
Angle = |30H - 5.5M|, where H is the hour and M is the minutes
Example: In a clock problem, to find the time when the hands form a right angle (90°), you would solve the equation: |30H - 5.5M| = 90°
The page also introduces the multinomial expansion formula for expressions like (Ax + By + Cz)ⁿ.
Vocabulary: Multinomial expansion extends the concept of binomial expansion to expressions with more than two terms.
This page continues with analytic geometry, covering triangles defined by coordinates and introducing conic sections.
For triangles defined by coordinates (x₁, y₁), (x₂, y₂), and (x₃, y₃), the following are presented:
Vocabulary: The centroid of a triangle is the point where its three medians intersect, dividing each median in a 2:1 ratio.
The page then introduces the general form of conic sections: Ax² + Bxy + Cy² + Dx + Ey + F = 0
When B = 0, the equation simplifies to: Ax² + Cy² + Dx + Ey + F = 0
Definition: Conic sections are curves formed by the intersection of a plane and a double cone. They include circles, ellipses, parabolas, and hyperbolas.
The type of conic section can be determined based on the coefficients:
Example: The equation x² + y² = 25 represents a circle with radius 5 centered at the origin.
Highlight: Understanding conic sections is crucial for many applications in physics, engineering, and advanced mathematics.
This page provides information on the properties of regular polyhedra, also known as Platonic solids, which are important in solid geometry.
The five Platonic solids are presented with their properties:
For each polyhedron, the following properties are listed:
Highlight: Euler's formula for polyhedra that do not intersect themselves: F + V - E = 2, where F is the number of faces, V is the number of vertices, and E is the number of edges.
Example: For a cube with side length s:
Vocabulary: A regular polyhedron is a three-dimensional solid where all faces are congruent regular polygons, and all vertices are surrounded by the same number of faces.
This page introduces fundamental concepts of analytic geometry, focusing on equations of lines and distance formulas.
Various forms of line equations are presented:
Definition: The slope (m) of a line is the change in y divided by the change in x between any two points on the line.
Distance formulas are provided for:
The page also includes formulas for parallel lines and the angle of inclination of a line.
Example: The distance between points (1, 2) and (4, 6) is: d = √[(4 - 1)² + (6 - 2)²] = √(3² + 4²) = 5
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Average App Rating
Students use Knowunity
In Education App Charts in 12 Countries
Students uploaded study notes
iOS User
Stefan S, iOS User
SuSSan, iOS User
