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: recognize the standard form of a quadratic equation, understand the derivation of the quadratic formula, solve quadratic equations using the quadratic formula.
Overview
- Standard Form Of A Quadratic Equation
- The Quadratic Formula
- Derivation Of The Quadratic Formula
We have observed that a quadratic equation is an equation of the form
where
is the coefficient of the quadratic term,
is the coefficient of the linear term, and
is the constant term.
The equation
is the
standard form of a quadratic equation.
Sample set a
Determine the values of
and
Practice set a
Determine the values of
and
in the following quadratic equations.
The solutions to
all quadratic equations depend only and completely on the values
and
When a quadratic equation is written in standard form so that the values
and
are readily determined, the equation can be solved using the
quadratic formula . The values that satisfy the equation are found by substituting the values
and
into the formula
Keep in mind that the plus or minus symbol,
is just a shorthand way of denoting the two possibilities:
The quadratic formula can be derived by using the method of completing the square.
Solve
for
by completing the square.
Now we have the proper form to complete the square. Take one half the coefficient of
square it, and add the result to both sides of the equation found in step 2.
(a)
is one half the coefficient of
(b)
is the square of one half the coefficient of
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Add the two fractions on the right side of the equation. The LCD
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Solve for
using the method of extraction of roots.
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Sample set b
Solve each of the following quadratic equations using the quadratic formula.
- Identify
and
- Write the quadratic formula.
- Substitute.
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- Identify
and
- Write the quadratic formula.
- Substitute.
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- Identify
and
- Write the quadratic formula.
- Substitute.
This equation has no real number solution since we have obtained a negative number under the radical sign.
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- Identify
and
- Write the quadratic formula.
- Substitute.
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- Write the equation in standard form.
- Identify
and
- Write the quadratic formula.
- Substitute.
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Practice set b
Solve each of the following quadratic equations using the quadratic formula.
Exercises
For the following problems, solve the equations using the quadratic formula.
Exercises for review