Understanding common algebraic equations is key for mastering College Algebra and acing the PSAT. These equations, like linear, quadratic, and exponential, help model real-world situations and solve problems across various fields, from science to finance.
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Linear equations (y = mx + b)
- Represents a straight line on a graph.
- 'm' is the slope, indicating the steepness and direction of the line.
- 'b' is the y-intercept, the point where the line crosses the y-axis.
- Can be solved for y or x to find specific points on the line.
- Used in various applications, including predicting trends and relationships.
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Quadratic equations (axยฒ + bx + c = 0)
- Forms a parabola when graphed, which can open upwards or downwards.
- Solutions can be found using factoring, completing the square, or the quadratic formula.
- The vertex represents the maximum or minimum point of the parabola.
- The discriminant (bยฒ - 4ac) determines the nature of the roots (real and distinct, real and repeated, or complex).
- Commonly used in physics, engineering, and finance.
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Exponential equations (y = aโฝแตหฃโพ)
- Represents growth or decay processes, such as population growth or radioactive decay.
- The base 'b' indicates the growth factor; if b > 1, itโs growth; if 0 < b < 1, itโs decay.
- The y-intercept is at (0, a), where 'a' is the initial value.
- Can be solved using logarithms to find the exponent.
- Important in fields like biology, economics, and computer science.
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Logarithmic equations (y = log_a(x))
- The inverse of exponential functions, used to solve for exponents.
- 'a' is the base, and the equation answers the question: "To what power must 'a' be raised to get 'x'?"
- Logarithmic scales are useful for measuring phenomena that span large ranges, like sound intensity (decibels) or earthquake magnitude (Richter scale).
- Properties include the product, quotient, and power rules, which simplify complex logarithmic expressions.
- Essential in data analysis and scientific calculations.
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Absolute value equations (|x| = a)
- Represents the distance of 'x' from zero on the number line.
- Has two possible solutions: x = a and x = -a, provided a โฅ 0.
- Graphically, it forms a V-shape on the coordinate plane.
- Used in real-world scenarios to express non-negative quantities, such as distance or error.
- Can be combined with other equations to solve more complex problems.
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Rational equations (algebraic fractions)
- Involves fractions where the numerator and/or denominator are polynomials.
- Can be solved by finding a common denominator or cross-multiplying.
- Important to identify restrictions on the variable to avoid division by zero.
- Graphically, they can have asymptotes where the function is undefined.
- Commonly used in rate problems, mixture problems, and real-world applications.
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Radical equations (square roots, cube roots)
- Involves expressions with roots, such as โx or โx.
- Solutions can introduce extraneous roots, so checking solutions is essential.
- Can be solved by isolating the radical and then squaring both sides (or cubing for cube roots).
- Graphically, they can have unique shapes and behaviors, especially near the origin.
- Useful in geometry, physics, and engineering contexts.
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Systems of linear equations
- Consists of two or more linear equations that share common variables.
- Can be solved using substitution, elimination, or graphing methods.
- Solutions can be unique (one intersection), infinite (coincident lines), or none (parallel lines).
- Applications include optimization problems and modeling real-world scenarios.
- Important in economics, engineering, and social sciences.
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Polynomial equations
- Involves expressions with variables raised to whole number powers.
- Can have multiple terms and degrees, affecting the shape of the graph.
- Solutions can be found using factoring, synthetic division, or the Rational Root Theorem.
- The Fundamental Theorem of Algebra states that a polynomial of degree n has n roots (real or complex).
- Widely used in various fields, including physics, engineering, and computer graphics.
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Inequalities and their graphs
- Represents a range of values rather than a specific solution.
- Can be linear or nonlinear, with solutions often depicted on a number line or coordinate plane.
- Important to understand how to graph inequalities, including shading regions and using dashed/solid lines.
- Can be combined to form systems of inequalities, which can have multiple solutions.
- Applications include optimization, economics, and decision-making processes.