How do I use the Quadratic Formula?

What does the Quadratic Formula help us find? What is it exactly? To put it simply, the quadratic formula is an algebraic formula that allows you to find the x-intercepts or roots of a quadratic equation.

What does the Quadratic Formula help us find?

The quadratic formula is a formula that allows you to find the x – intercepts or roots of a quadratic equation.

A quadratic equation can have three different root(s) outcomes.  

1.    Two rational or two irrational roots where the equation hits the x- axis in two spots.

Quadratic Formula w one rational or irrational root

2.    One rational or one irrational root where the equation hits the x-axis in one spot.

3.    Two complex roots where the equation does nothit the x-axis at all.

Quadratic Formula w two complex roots not hitting x-axis

How do I use the Quadratic Formula?

Now that you are familiar with the types of roots a quadratic equation can produce, let us look at the quadratic formula.

The Quadratic Formula

Quadratic Formula Equation

To effectively use the Quadratic Formula, we must know the standard form of a quadratic. This will be used to help you identify the values of a, b, and c.

Standard Form of a Quadratic

Standard form of a quadratic

Quadratic Formula Problem Example 1:

Use the Quadratic Formula to solve,

Example Quadatic Formula Problem

Step 1: Make sure your equation is in standard form.Our equation is already in standard form.

Step 2: Identify your a,b, and c from your equation by looking at the general standard form equation.

a = 1, b = 6, and c = -16

Step 3: Plug in your values from step 2 into the Quadratic Formula.

Quadratic Formula - Problem Set

Step 4: Simplify - Solve the Quadratic Formula!

So, the roots are at x = 2 and x = -8. There are two rational roots because the roots are rational numbers.

Quadratic Formula Problem Example 2:

Use the Quadratic Formula to solve,

Example Quadratic Formula Problem 2

Step 1: Make sure your equation is in standard form.The equation is not in standard form, you will need to add ten to both sides to look like this.

Quadratic Formula is standard form example

Step 2: Identify your a, b, and c from your equation by looking at the general standard form equation.

a = 1, b = -6, and c = 10

Step 3: Plug in your values from step 2 into the Quadratic Formula.

Quadratic Formula - Plug in Values example 2

Step 4: Simplify - Solve using the Quadratic Formula!

So, the roots are at x= 3 + i and x = 3 - i, which are both complex roots because you can’t take the square root of a negative number. Thus, in the equation, the parabola does not hit the x-axis.

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