Factoring to Solve Quadratic Equations

Topics for Today:

All of the work we have done on factoring has led to today's topic of solving quadratic equations by factoring.  We talked about the zero-product property (when multiplying, if one factor is zero, then the equation equals zero), and how we use it to find our solutions (also called roots or zeroes). 

An example of an equation that requires several steps to solve is included here:

Vocabulary:  zero-product property

Sections Covered in Textbook:

10-5: Factoring to Solve Quadratic Equations (pages 536-540)

Resources & Tutorials:

1) What is the zero-product property?

2) How do you use the zero product property to find solutions to an equation?

3) How do you solve a quadratic equation by factoring?

Solving Quadratic Equations

Topics for Today:

Solving quadratic equations was the topic of the day.  We solved these equations by graphing and by using algebra.  For quadratic equations, we have three possibilities for our solutions:  we may have two solutions, one solution, or no REAL solutions.  The rules of algebra still apply when solving numerically - whatever we do to one side of the equation, must also be done to the other side to keep the truth of the equals sign.  Students were also reminded that squaring and taking the square root are inverse operations. 

Sections Covered in Textbook:

10-4: Solving Quadratic Equations (pages 529-534)

Resources & Tutorials:

1)  How do you solve a quadratic equation with two solutions by graphing?

2)  How do you use the square root method to solve a quadratic equation with two solutions?

Finding and Estimating Square Roots

Topics for Today:

Today we discussed perfect squares and square roots.  Squaring and taking the square root are inverse operations.  Students will be asked to memorize the common perfect squares, and there is a Quizlet set that should hopefully make learning them fun.

Vocabulary: square root, principal square root, negative square root, radical, radicand, perfect squares

Sections Covered in Textbook:

10-3: Finding and Estimating Square (pages 524-528)

Resources & Tutorials:

1) What is a perfect square?
2) How do you find the square root of a perfect square?
3) How do you find the square root of a fraction?
4) How do you estimate a square root of a non-perfect square?

Quadratic Functions

Topics for Today:

Quadratic functions are still the topic of the day.  Today we worked with the axis of symmetry and used it to find our vertex.  Because parabolas are symmetric, we are able to find points on one side of the axis of symmetry and reflect them to the other side of the axis of symmetry.  Once we have the vertex, and a few points on either side of the axis of symmetry, we can easily draw our parabola.

Sections Covered in Textbook:

10-2: Quadratic Functions (pages 517-523)

Resources & Tutorials:

1) How do you find the axis of symmetry?
2) Find the axis of symmetry and your vertex

Exploring Quadratic Graphs

Topics for Today:

Today we began our work on quadratic functions.  Quadratic functions, simply stated, are functions that have a variable with the highest degree exactly equal to two.  We looked at the standard form of a quadratic function and looked at graphs of different parabolas.

Vocabulary: quadratic function, standard form of a quadratic function, parabola, axis of symmetry, vertex, minimum, maximum

Sections Covered in Textbook:

10-1: Exploring Quadratic Graphs (pages 510-516)

Resources & Tutorials:

1) What is a quadratic function?
2) What is a parabola?

Factoring Trinomial Special Cases

Topics for Today:

We are back to pattern recognition for factoring.  When we multiplied binomials by squaring them or by multiplying a difference, we noted a pattern for the resulting products.  Today, we worked backward from the trinomial (in the case of perfect square trinomials) or the binomial (in the case of difference of squares) to determine the two binomial factors. 

Students are reminded that now would be a good time to memorize the common perfect squares.  We also talked about square roots in the context of being the opposite of squaring numbers.  We'll deal with radicals a little later on, in May. 

For perfect square trinomials, students should be asking the questions:

• Is the first variable term a perfect square?
• If yes, is the constant term a perfect square?
• If yes, is the middle term equal to two times the square roots of the first and third terms?

What about factoring difference of squares?  We have another pattern to follow for this type of polynomial.  For the difference of squares, students should be asking the questions:

• Is the variable piece a perfect square?
• Is the constant piece a perfect square?
• Is the operation being performed subtraction?

Sections Covered in Textbook:

9-7: Factoring Special Cases

Resources & Tutorials:

1) How to use a shortcut to factor a perfect square trinomial
2) How do you factor using the difference of squares

Factoring Trinomials Part 2 - Split the Middle (a>1)

Topics for Today:

We expanded our discussion today to include factoring polynomials where the leading coefficient is not 1.  We used the product-sum game to work with factors so we could "split the middle" of the equation, and then factor by grouping.

Sections Covered in Textbook:

9-6: Factoring Trinomials of the type ax² + bx + c (where a ≠ 1)

       (pages 486-489)

Resources & Tutorials:

1) Factor a trinomial using A-C method
    (This is a different method from what was introduced in class.)
2) Factor a trinomial with a > 1
    (This method is more like what was introduced in class.)

3) Factoring Using the Box Method
    (This is yet another method - used in many high school Alg II classes)