Common Algebraic Equations

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 (ax² + 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 (b² - 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 = a⁽ᵇˣ⁾)

   • Represents growth or decay processes, such as population growth or radioactive decay.
   • The base 'b' indicates the growth factor; if b > 1, it’s growth; if 0 < b < 1, it’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 ≥ 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 √x or ∛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.