These important formulas are essential tools for solving various mathematical problems. From geometry to algebra, understanding these concepts will help you tackle equations, calculate areas, and analyze relationships, making your study guides more effective and comprehensive.
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Quadratic Formula
- Used to find the roots of a quadratic equation in the form ax² + bx + c = 0.
- The formula is x = (-b ± √(b² - 4ac)) / (2a).
- The discriminant (b² - 4ac) determines the nature of the roots: two real, one real, or two complex.
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Pythagorean Theorem
- Relates the lengths of the sides of a right triangle: a² + b² = c².
- 'c' represents the hypotenuse, while 'a' and 'b' are the other two sides.
- Useful for calculating distances and verifying right angles in geometric problems.
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Area of a Circle
- The area is calculated using the formula A = πr², where 'r' is the radius.
- Important for problems involving circles in geometry and real-world applications.
- π (pi) is approximately 3.14, but can be used as a fraction (22/7) for simpler calculations.
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Slope-Intercept Form
- The equation of a line is expressed as y = mx + b, where 'm' is the slope and 'b' is the y-intercept.
- Slope indicates the steepness and direction of the line.
- Useful for graphing linear equations and understanding relationships between variables.
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Distance Formula
- Calculates the distance between two points (x₁, y₁) and (x₂, y₂) using the formula: d = √((x₂ - x₁)² + (y₂ - y₁)²).
- Essential for coordinate geometry and real-world distance problems.
- Derives from the Pythagorean theorem.
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Circumference of a Circle
- The circumference is calculated using the formula C = 2πr or C = πd, where 'd' is the diameter.
- Represents the distance around the circle.
- Important for problems involving circular objects and motion.
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Volume of a Cylinder
- The volume is calculated using the formula V = πr²h, where 'r' is the radius and 'h' is the height.
- Useful for determining capacity in real-world applications.
- Relates to the area of the base (circle) and the height of the cylinder.
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Exponential Growth/Decay
- Describes processes that increase or decrease at a rate proportional to their current value.
- The general formula is y = a(1 ± r)ⁿ, where 'a' is the initial amount, 'r' is the rate, and 'n' is time.
- Commonly used in finance, biology, and physics to model population growth or radioactive decay.
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Trigonometric Ratios (sine, cosine, tangent)
- Ratios defined for a right triangle:
- Sine (sin) = opposite/hypotenuse
- Cosine (cos) = adjacent/hypotenuse
- Tangent (tan) = opposite/adjacent.
- Fundamental in solving problems involving angles and distances in triangles.
- Basis for more advanced topics in trigonometry and calculus.
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Logarithmic Formula
- The logarithm is the inverse operation of exponentiation, expressed as log_b(a) = c, meaning b^c = a.
- Common bases include 10 (common logarithm) and e (natural logarithm).
- Useful in solving exponential equations and in applications involving growth rates and scales.