how to solve a non-factorable quadratic congruence

TL;DR

Learn two methods for solving quadratic congruences mod 7.

Transcript

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Key Insights

• 😑 Quadratic congruences mod 7 can be solved by using the quadratic formula or by factoring the quadratic expression.
• 😑 Factoring the quadratic expression requires manipulating the coefficients and constants to make the expression factorable.
• 📳 The solutions to the quadratic congruences mod 7 may not always be integers, but they can be found using modular arithmetic.
• 📳 The solutions to the quadratic congruences mod 7 can be represented as X congruent to a certain value mod 7.
• 😑 The factorability of the quadratic expression can depend on whether the constant term can be manipulated to make it factorable.
• 💼 It is important to consider whether the modulus (in this case, 7) is prime or not to determine the solvability of the quadratic congruences.
• 😄 The use of the quadratic formula or factoring depends on the nature of the quadratic expression and the ease of finding solutions.

Questions & Answers

Q: What is a quadratic congruence mod 7?

A quadratic congruence mod 7 is an equation where the unknown variable is raised to the power of 2 and is congruent to a certain value mod 7.

Q: What is the quadratic formula?

The quadratic formula is used to solve quadratic equations and is given by x = (-b ± √(b^2 - 4ac))/2a, where a, b, and c are the coefficients of the quadratic equation.

Q: How do you factor a quadratic expression?

To factor a quadratic expression, you need to find two binomials that multiply together to give the quadratic expression.

Q: Why is it important to consider whether 7 is prime or not in solving quadratic congruences?

The fact that 7 is prime allows for certain manipulations of the equations and simplifications to find the solutions more easily.

Summary & Key Takeaways

• The video discusses two methods for solving quadratic congruences mod 7: using the quadratic formula and factoring the quadratic expression.

• The quadratic formula method involves computing the discriminant and using the square root to find the solutions.

• The factoring method involves manipulating the equation to make it factorable, then solving for the solutions.