LINEAR-EQUATION (2)
Linear Equations in One Variable
Definition: A linear equation in one variable can be expressed in the standard form:
Equation: ax + b = 0
Components:
a: non-zero coefficient of x (the slope)
b: constant term (the y-intercept)
Graphing: Represents a straight line on the coordinate plane.
Solution: The value of x that satisfies the equation can be found by isolating x.
Quadratic Equations in One Variable
Definition: A quadratic equation in one variable can be expressed in the standard form:
Equation: ax² + bx + c = 0
Components:
x: variable
a, b, c: constants (a ≠ 0)
Graphing: The graph is a parabola.
Parabola opens upward if a > 0 and downward if a < 0.
Solutions: Can have 0, 1, or 2 real solutions depending on the discriminant (D = b² - 4ac).
Quadratic Formula: x = (-b ± √D) / 2a
Page 6: Equations Containing Rational Expressions
Definition: Involves fractions where the numerator and/or denominator is a polynomial.
General Form: P(x)/Q(x) = R(x)
P(x), Q(x): polynomials (Q(x) ≠ 0)
R(x): polynomial or rational expression
Page 7: Examples of Rational Expressions
Example 1: (x + 4)(x - 2) = 3
Example 2: 5(x + 3) - 2(x - 1) = 0
Application: Tank problem: Drains at rate of 100 L/hour; find time to empty 200 L.
Page 8: Equations Containing Radicals of Index 2
Definition: Equations with variables under a square root.
General Form: √f(x) = g(x),
Where f(x) and g(x) involve x, keeping the square root real.
Page 9: Examples of Radical Equations
No specific examples provided on this page.
Page 10: Equations in Quadratic Form
Definition: Equations that can be rewritten as a quadratic by substituting a new variable.
General Form: au² + bu + c = 0
where u is an expression involving the original variable x.
Solving: Solve for u as if it were quadratic, then substitute back to find x.
Page 11: Examples of Quadratic Form Equations
Example 1: Solve x⁴ - 5x² + 4 = 0
Example 2: Solve (2x)^(4/3) - 7x^(2/3) + 3 = 0
Example 3: Solve x^(2/3) - 5(1/3) + 6 = 0
Example 4: Solve x⁶ - 5x³ + 4 = 0
Example 5: Solve x - 2 - 5x - 1 - 6 = 0
Page 12: Polynomial Equations (Degree 3 or More)
Definition: Involves terms where the variable is raised to powers of 3 or higher.
General Form: axⁿ + ... + a₀ = 0 (where n > 3)
Coefficients ai are constants (an ≠ 0)
Page 13: Examples of Cubic and Higher-Degree Polynomials
Example 1: Solve x³ - 6x² + 11x - 6 = 0
Example 2: Solve 4x - 2x³ - 7x² + 8x + 12 = 0
Page 14: Polynomial Equations (Degree 3 or More)
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