16 Possible values for t

TL;DR

The video explains how to solve equations of the form x + 1/x = t, determining the possible values of t.

Transcript

• [Voiceover] Let inverted, or upside down triangle x, be defined as x plus one over x. Let me write that down. Inverted triangle x, is equal to x plus one over x. For all non-zero integers x, all right, if inverted or upside down triangle x is equal to t. So if this whole thing is equal to t, where t is an integer, which of the following is a poss... Read More

Key Insights

• ☺️ Inverted Triangle x is defined as x + 1/x.
• 😰 The equation x + 1/x = t can be rearranged as a quadratic equation to find x.
• ☺️ Factoring the quadratic equation helps determine possible values of x for each t.
• 😰 Of the given options, only t = -2 has a non-zero integer solution, x = -1.

Questions & Answers

Q: How can the equation x + 1/x = t be solved for x?

By rearranging the equation as x^2 - tx + 1 = 0, you can factor it to find the possible values of x for each t.

Q: Which value of t results in a non-zero integer solution, x?

The only value of t that leads to a non-zero integer solution is -2. This results in x = -1.

Q: Can all the equation choices be factored to obtain integer solutions for x?

No, only the equation x^2 - x + 1 (t = 1) can be factored as (x + 1)^2 = 0, resulting in x = -1. The other choices cannot be factored to obtain integer solutions for x.

Q: Why is t = -2 the only value with a solution?

When t = -2, the equation simplifies to (x + 1)^2 = 0, and x = -1. For the other values of t, no two integers can be found to satisfy the factoring of the quadratic equation.

Summary & Key Takeaways

• The video introduces the equation x + 1/x = t, where x is a non-zero integer and t is an integer.

• To find possible values of t, rearrange the equation to solve for x as a function of t.

• By analyzing the possible values of x for each t, you can determine that only t = -2 results in a non-zero integer solution, x = -1.