Name: ___________________________ Per: ____ Date: ___________
Investigation 9.7 – The Quadratic Formula – Deriving the Quadratic Formula
You have learned several methods for solving quadratic equations symbolically. Completing the square is particularly
useful because it can be used for any quadratic equation. But if your equation is something like
, completing the square will be very messy! In this lesson you’ll find a formula that solves any
quadratic equation in the form
quadratic formula.
directly, using only the values of a, b, and c. It is called the
Recall that you can solve some quadratic equations symbolically by recognizing their forms:
You can also undo the order of operations in other quadratic equations when there is no x-term, as in these:
If the quadratic expression is in the form
a rectangle diagram.
you can complete the square by using
In this investigation you’ll use the completing the square method to derive the quadratic
formula.
Investigation: Deriving the Quadratic Formula
You’ll solve
and develop the quadratic formula for the general case in the process.
Step 1: Identify the values of a, b, and c in the general form
for the equation
.
a=
b=
c=
Step 2: Group all the variable terms on the left side of your equation so that it is in the form:
Step 3: It’s easiest to complete the square when the coefficient of x2 is 1. So divide your equation by the value of a.
Write it in the form:
Step 4: Use a rectangle diagram to help you complete the square.
What number must you add to both sides?
Write your new equation in the form:
Step 5: Rewrite the trinomial on the left side of your equation as a squared binomial. On the right side, find a common
denominator. Write the next stage of your equation in the form:
Step 6: Take the square root of both sides of your equation, like this:
Step 7: Rewrite
Then get x by itself on the left side, like this:
Step 8: There are two possible solutions given by the equations:
Write your two solutions in radical form.
Step 9: Write your solutions in decimal form.
Step 10: Consider the expression:
real numbers?
What restrictions should there be so that the solutions exist and are
The quadratic formula gives the same solutions that completing the square or factoring does. You don’t need to derive
the formula each time. All you need to know are the values of a, b, and c. Then you substitute these values into the
formula.
In the next example you’ll learn how to use the formula for quadratic equations in general form. You can even use it
when the values of a, b, and c are decimals or fractions.
Example:
Use the quadratic formula to solve 3x2 + 5x – 7 = 0.
The equation is already in general form, so identify the values of a, b, and c.
a=
b=
c=
Then, here is one way to use the formula:
The two exact roots of the equation are
.
Calculate the roots in decimal form:
To make the formula simpler, think of the expression under the square root sign as one number. This
expression,
d=
, is called the discriminant. In the example, the discriminant is 109. So let
Then the formula becomes:
Homework: DA 9.7 Pg. 533 #1-3, 5-7, 9-12