This module is from Elementary Algebra</link> by Denny Burzynski and Wade Ellis, Jr.
Methods of solving quadratic equations as well as the logic underlying each method are discussed. Factoring, extraction of roots, completing the square, and the quadratic formula are carefully developed. The zero-factor property of real numbers is reintroduced. The chapter also includes graphs of quadratic equations based on the standard parabola, y = x^2, and applied problems from the areas of manufacturing, population, physics, geometry, mathematics (numbers and volumes), and astronomy, which are solved using the five-step method.Objectives of this module: be able to solve quadratic equations using the method of extraction of roots, be able to determine the nature of the solutions to a quadratic equation.
Overview
- The Method Of Extraction Of Roots
- The Nature Of Solutions
Quadratic equations of the form
can be solved by
the method of extraction of roots by rewriting it in the form
To solve
we are required to find some number,
that when squared produces
This number,
must be a square root of
If
is greater than zero, we know that it possesses two square roots,
and
We also know that
We now have two replacements for
that produce true statements when substituted into the equation. Thus,
and
are both solutions to
We use the notation
to denote both the principal and the secondary square roots.
The nature of solutions
Solutions of
For quadratic equations of the form
- If
is greater than or equal to zero, the solutions are
- If
is negative, no real number solutions exist.
- If
is zero, the only solution is 0.
Sample set a
Solve each of the following quadratic equations using
the method of extraction of roots.
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Calculator problem. Solve
Round to the nearest hundredth.
Rounding to the nearest hundredth produces 4.10. We must be sure to insert the
symbol.
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Practice set a
Solve each of the following quadratic equations using the method of extraction of roots.
Sample set b
Solve each of the following quadratic equations using the method of extraction of roots.
Practice set b
Solve each of the following quadratic equations using the method of extraction of roots.
Exercises
For the following problems, solve each of the quadratic equations using the method of extraction of roots.
For the following problems, solve for the indicated variable.
For the following problems, solve each of the quadratic equations using the method of extraction of roots.
calculator problems
For the following problems, round each result to the nearest hundredth.
Exercises for review