QF - Factoring and Solving Quadratic Equations (Overview)

Factoring and Solving Quadratic Equations

Introduction

Now that you have learned how to solve, interpret, create and manipulate linear functions, you can move on to Quadratic Functions. Let's sa you are a contractor and you are working on the backyard. You decide to plant a vegetable garden, and you want it to take 24 square feet of planting space. You also want one side to be 2 feet longer than the other side. How would you figure out what dimensions to make your garden? It would take forever to try every combination - so that is why algebra and in particular quadratic equations are useful!

Essential Questions

• How is a relation determined to be quadratic?
• How do I choose the most efficient method of solving quadratic equations?
• How do the factors of a quadratic function yield the zeros for that function?
• How is the quadratic formula developed by completing the square?
• How can the quadratic formula be used to find the zeros of a quadratic function?
• Under what circumstances can one take the square root of both sides of the equation?
• What are the relative advantages and disadvantages of solving a quadratic function by factoring, completing the square, quadratic formula, or taking the square root of both sides?
• How do I interpret quadratic functions in context?

Key Terms

Complete factorization over the integers - Writing a polynomial as a product of polynomials so that none of the factors is the number 1, there is at most one factor of degree zero, each polynomial factor has degree less than or equal to the degree of the product polynomial, each polynomial factor has all integer coefficients, and none of the factors of the polynomial can be written as such a product.

Complete the Square - The process of converting a quadratic equation into a perfect square trinomial by adding or subtracting terms on both sides.

Difference of Two Squares - A squared (multiplied by itself) number subtracted from another squared number. It refers to the identity a^2 −b^ 2 = (a + b)(a − b) in elementary algebra.

Factor - The opposite operation of distributing. 

Perfect Square Trinomial - A trinomial that factors into two identical binomial factors.

Quadratic Equation - An equation of degree 2, which has at most two solutions.

Quadratic Function - A function of degree 2 which has a graph that "turns around" once, resembling an umbrella-like curve that faces either right-side-up or upside down. This graph is called a parabola.

Root - The x-values where the function has a value of zero.

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