factoring a quadratic by completing the square

TL;DR

Learn how to factor a quadratic expression with complex numbers using completing the square.

Transcript

[doorbell sound] Okay Sean, this video's for you. I'll show you how to factor x² +3x + 4 and we know that we can factor this in real numbers— We're going to use complex numbers here. And I'll do this with completing the square. First of all we notice that we have a one in front of the x² so that's good So you see that this is going to be x² + 3x an... Read More

Key Insights

• 😑 Factoring a quadratic expression with complex numbers is possible through completing the square.
• 💯 The concept of perfect square trinomials is crucial in the factoring process.
• 😑 Converting an expression into a difference of two squares allows for easier factorization.

Questions & Answers

Q: What method is used to factor the quadratic expression x² + 3x + 4 with complex numbers?

The method used is completing the square, where the quadratic expression is manipulated to create a perfect square trinomial and then converted into a difference of two squares.

Q: How is the magic number determined in completing the square?

The magic number is calculated by taking half of the coefficient of the x term, squaring it, and adding it to the expression. In this case, the magic number is 9/4.

Q: Why is the plus sign changed to a minus sign when converting the expression into a difference of two squares?

The plus sign is changed to a minus sign to create a perfect square trinomial inside the parentheses. This is achieved by using the fact that i² is equal to -1.

Q: How is the final factored form of the quadratic expression written?

The final factored form is (3 - i√7)/2 multiplied by (x + 3 + i√7)/2. Both parts of the expression have a common denominator.

Summary & Key Takeaways

• The video explains how to factor the quadratic expression x² + 3x + 4 using complex numbers and completing the square.

• The process involves finding the perfect square of the first three terms, adding and subtracting a magic number, and converting the expression into a difference of two squares.

• The final factored form is (3 - i√7)/2 multiplied by (x + 3 + i√7)/2.