Learning Quadratic Equations

We will go over how to solve for quadratic functions by factoring, completing the square, and using the quadratic formula.

Factoring Quadratics

We factor polynomials in order to use Zero Product Property which is how we find the x-intercept(s) of a quadratic function. You can factor by area models, or by the grouping method.

For both of these methods you need to find two numbers that multiply to the same as the coefficient (number before a variable) of the squared variable multiplied with the constant (number by itself). Those same two numbers have to also add up to the coefficient of the variable to the first power in that expression. ...

To find these two numbers, we use a diamond puzzle.

Quadratic Standard Form:

ax²+bx+c

The diamond puzzle puts the product (answer of two numbers multiplied together) of the "a" and "c" value at the top of the diamond, and the b value at the bottom of the diamond.

Area Models

For factoring using the area model method, you will make a 2x2 box. In the bottom right square you will put the third term of your expression in standard form, in the top left you will put the first term, and in the remaining two squares you will put one number we found from the diamond puzzle in each box. You will then solve for the values in the box by putting numbers on the outside of the box (two sides) that will multiply to equal each of the values inside the box. On each of the sides of the box where you put numbers that multiplied to the box's values, you will find part of your original expression's factored form. You will write the factored form of the expression as a product.

Grouping

For the grouping method, you will first see if your expression (which is in standard form) has a Greatest Common Factor (GCF), which you will then separate from the expression with parentheses. You will put the numbers you found with the diamond puzzle as a replacement for the second term of your expression. Now you should have four separate terms in your expression, and you will then find the GCF of the first two terms, and the second two terms, while still having the original addition/subtraction sign between these two sets of terms. You will take the GCFs of each set of terms as one factor, and the expression left after the GCFs (each set of terms should have the same expression) ...

as the second factor for your original expression. You will write factored form as a product (two factors multiplied together), just as you did using the area model method of factoring.

After using either factoring method, you will use Zero Product Property to find the x-intercept(s) of your quadratic function. For Zero Product Property, you set each factor in your factored expression equal to 0 and then solve for the x-intercept(s).

Example Using Factoring:

Completing the Square