Algebra II Solving Equations Flashcards

Solving Equations Overview

• Various equations produce specific solutions for the variable x, showcasing different mathematical concepts and operations.

Absolute Value Equations

• Equation: 3|x - 7| + 4 = 10

• Solutions:* x = 5, x = 9
  Indicates how absolute values affect the potential solutions by isolating x.

• Equation: -2|x + 4| = 12

• Solution:* No solution
  Demonstrates that negative coefficients with absolute values can invalidate solutions.

• Equation: 5 + |2x - 1| = 8

• Solutions:* x = -1, x = 2
  Shows how solving absolute value equations includes finding both positive and negative cases.

Quadratic Equations

• Equation: x² + x - 6 = 0

• Solutions:* x = -3, x = 2
  Quadratic equations can have two real solutions, determined by factorization.

• Equation: 3x² - 8 = 10x

• Solutions:* x = -2/3, x = 4
  Involves rearranging the equation before applying the quadratic formula or factoring.

• Equation: x² = -2x - 1

• Solution:* x = -1
  Simplifying to standard form reveals a straightforward solution.

• Equation: x⁴ + 5x² - 36 = 0

• Solutions:* x = ±3i, x = ±2
  Illustrates how higher degree polynomials can result in imaginary and real solutions.

Higher-Degree Polynomial Equations

• Equation: 2x³ - 2x² - 40x = 0
• Solutions:* x = -4, x = 0, x = 5
  Factoring out common terms leads to straightforward solutions for cubic equations.

Rational and Radical Equations

• Equation: (x + 5)/(x - 2) = 7/3

• Solution:* x = 29/4
  Rational equations require cross-multiplication to simplify the equation.

• Equation: √(5x + 2) - 3 = 1

• Solution:* x = 14/5
  Shows how squaring both sides of a radical equation can isolate x.

• Equation: 2√(4x - 1) + 7 = 13

• Solution:* x = 5/2
  Highlights manipulation of radicals and balancing equations.

Miscellaneous Equations

• Equation: ³√(x - 5) - 2 = 0

• Solution:* x = 13
  Solving involves isolating the cube root and cubing both sides.

• Equation: 5 + 3⁴√(2x) = 11

• Solution:* x = 8
  Reflects an application of root operations to isolate x.

• Equation: -3/x² + 2/x = 1/3

• Solution:* x = 3
  Involves common denominators and careful algebraic manipulation.

• Equation: 2x/(x² + 5x + 4) + 3/(x + 1) = 6/(x + 4)

• Solution:* x = 6
  Requires finding common denominators and algebraic restructuring to solve.

• Equation: x³ - 8x² + 16x = 0

• Solutions:* x = 0, x = 4
  Factors into simpler linear components, revealing solutions directly.

• Equation: x²/8 + 1 = 3

• Solutions:* x = ±4
  Demonstrates how to manipulate equations involving fractions to find values of x.

• Equation: |3x - 12| + 16 = 16

• Solution:* x = 4
  Shows the effect of isolating absolute values to find solutions.