Common Algebraic Equations to Know for College Algebra

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.

  1. 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.
  2. 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.
  3. Exponential equations (y = aโฝแต‡หฃโพ)

    • Represents growth or decay processes, such as population growth or radioactive decay.
    • The base 'b' indicates the growth factor; if b > 1, itโ€™s growth; if 0 < b < 1, itโ€™s 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.
  4. 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.
  5. 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 โ‰ฅ 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.
  6. 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.
  7. Radical equations (square roots, cube roots)

    • Involves expressions with roots, such as โˆšx or โˆ›x.
    • 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.
  8. 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.
  9. 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.
  10. 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.


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ยฉ 2024 Fiveable Inc. All rights reserved.
APยฎ and SATยฎ are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.