Solving Quadratic Functions by Factoring

Today we take the final step with our factoring unit. Until now, we have been rewriting expressions; today, we use that skill to actually solve quadratic functions. This is where all your practice pays off!

The Key Concept: The Zero Product Property

To solve these equations, we rely on a simple but powerful rule: if two numbers multiplied together equal zero, one of them must be zero.

For example, if we have the equation:

$$ (x-3)(x+7) = 0 $$

Then either:

$$ x-3 = 0 \quad \text{or} \quad x+7 = 0 $$

Solving these simple linear equations gives us our answers: $x=3$ and $x=-7$.

Examples from Class Notes

As seen in the attached class notes, we looked at a few different scenarios. Here is a breakdown of the techniques we covered:

  • Trinomials: For $x^2 + 4x - 21 = 0$, we find factors of $-21$ that add up to $4$. $$(x+7)(x-3) = 0$$ Solutions: $x = -7, x = 3$
  • Greatest Common Factor (GCF): Always check for a GCF first! For $20x^2 - 15x = 0$, factor out $5x$: $$5x(4x-3) = 0$$ Set both parts to zero: $5x=0$ or $4x-3=0$. Solutions: $x = 0, x = \frac{3}{4}$
  • Difference of Squares: For $x^2 - 9 = 0$: $$(x-3)(x+3) = 0$$ Solutions: $x = 3, x = -3$

Homework Assignment

Please complete the following to practice these concepts. Use the attached answer sheet to check your work as you go. If you get stuck, look closely at the examples in the PDF notes.

  • Worksheet (Practice B): Problems #34, 35, and 36 (Solving equations).
  • Textbook (Pg. 82): Problems #37-45 all (Find the zeros).

Discussion Question: Snow Day Protocol

With the first snow falling, let's talk about Snow Days. When school is cancelled, it interrupts instruction and can complicate testing schedules. I want to make sure you are supported even when we aren't in the classroom.

Please reply with your preferences:

  1. Content: What helps you learn best from home? (e.g., A full lesson video recorded by me, links to explanation websites, or just the notes?)
  2. Assignments: What format do you prefer for snow day work? (e.g., A printable worksheet, a textbook assignment, or an online quiz?)

You will have assignments on snow days, so this is your chance to help shape what they look like!