Algebra II Solving Equations Flashcards

Questions and Answers

What are the solutions for the equation $3|x - 7| + 4 = 10$?

x = 5, x = 9

What is the solution for the equation $-2|x + 4| = 12$?

no solution

What are the solutions for the equation $5 + |2x - 1| = 8$?

x = -1, x = 2

What are the solutions for the equation $x^2 + x - 6 = 0$?

x = -3, x = 2 Signup and view all the answers

What are the solutions for the equation $3x^2 - 8 = 10x$?

x = -2/3, x = 4 Signup and view all the answers

What is the solution for the equation $x^2 = -2x - 1$?

x = -1 Signup and view all the answers

What are the solutions for the equation $3x(x + 1) - x^2 = -4x - 3$?

x = -1/2, x = -3 Signup and view all the answers

What is the solution for the equation $\sqrt{5x + 2} - 3 = 1$?

x = 14/5 Signup and view all the answers

What is the solution for the equation $2\sqrt{4x - 1} + 7 = 13$?

x = 5/2 Signup and view all the answers

What is the solution for the equation $³\sqrt{x - 5} - 2 = 0$?

x = 13 Signup and view all the answers

What is the solution for the equation $5 + 3⁴\sqrt{2x} = 11$?

x = 8 Signup and view all the answers

What is the solution for the equation $(x + 5)/(x - 2) = 7/3$?

x = 29/4 Signup and view all the answers

What are the solutions for the equation $(x + 4)/3 = 2/(x - 1)$?

x = -5, x = 2 Signup and view all the answers

What is the solution for the equation $-3/x² + 2/x = 1/3$?

x = 3 Signup and view all the answers

What is the solution for the equation $2x/(x^2 + 5x + 4) + 3/(x + 1) = 6/(x + 4)$?

x = 6 Signup and view all the answers

What are the solutions for the equation $2x^3 - 2x^2 - 40x = 0$?

x = -4, x = 0, x = 5 Signup and view all the answers

What are the solutions for the equation $x^4 + 5x^2 - 36 = 0$?

x = ±3i, x = ±2 Signup and view all the answers

What are the solutions for the equation $x^3 - 8x^2 + 16x = 0$?

x = 0, x = 4 Signup and view all the answers

What are the solutions for the equation $x^2/8 + 1 = 3$?

x = ±4 Signup and view all the answers

What is the solution for the equation $|3x - 12| + 16 = 16$?

x = 4 Signup and view all the answers

Study Notes

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.

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Test your knowledge of solving equations with these flashcards from Algebra II. Each card presents an equation or scenario that challenges your understanding of different solutions. Perfect for quick reviews or studying for exams!