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Solving Quadratic Equations: Completing the Square and the Quadratic Formula, Exams of Algebra

An in-depth exploration of solving quadratic equations through completing the square and the quadratic formula. Students will learn how to transform equations into the required format, identify constants, and apply the quadratic formula to find solutions. The document also introduces complex numbers as solutions to quadratic equations.

Typology: Exams

Pre 2010

Uploaded on 08/19/2009

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Ch. 9 Test Review (9.1-9.5) Math 102
Chapter 9 is about solving quadratic equations. A quadratic equation is an equation for which
the highest power of the variable is 2, and any quadratic equation can be written in the form:
2
0
ax bx c
+ + =
We learned how to solve these equations by factoring in ch. 6, but not all quadratic equations
can be solved with this technique. In ch. 9, we learn to solve by two new methods: completing
the square and the quadratic formula. New types of solutions are discovered – some of which
aren’t even “real” numbers! Note that there are always 2 solutions to a quadratic equation.
9.1: This section was preparation for completing the square in section 9.2. The main idea is
that if you have an expression of the form:
( )
2
variable expression constant
=
then you can write down two solutions:
variable expression = + constant
and
variable expression = - constant
Symbolically, the rule was summarized as:
If 2
x a
=
, then
x a
= ± .
You should note that the equation may have to be modified to fit this form before you
“take the square root of both sides”. The left hand side must be only the square of a
variable expression!
Note that if a quadratic equation is presented to you in this format, taking the square root
of both sides is by far the most convenient solution method.
9.2: Completing the square
The point of completing the square is to force any quadratic equation into the format in
9.1 above. This is done by simple algebra tricks, but the process can still be very
confusing:
1. Get all variables on one side, constants on the other.
2. Get a coefficient of 1 on the quadratic term.
3. Guess the binomial you square to get all the same variable parts.
4. Compensate for the constant you’ve implicitly added to the variable side.
pf3

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Ch. 9 Test Review (9.1-9.5) Math 102

Chapter 9 is about solving quadratic equations. A quadratic equation is an equation for which the highest power of the variable is 2, and any quadratic equation can be written in the form:

ax^2 + bx + c = 0

We learned how to solve these equations by factoring in ch. 6, but not all quadratic equations can be solved with this technique. In ch. 9, we learn to solve by two new methods: completing the square and the quadratic formula. New types of solutions are discovered – some of which aren’t even “real” numbers! Note that there are always 2 solutions to a quadratic equation.

9.1: This section was preparation for completing the square in section 9.2. The main idea is that if you have an expression of the form:

variable expression 2 =constant

then you can write down two solutions:

variable expression = + constant and variable expression = - constant

Symbolically, the rule was summarized as:

If x^2 = a , then x = ± a.

You should note that the equation may have to be modified to fit this form before you “take the square root of both sides”. The left hand side must be only the square of a variable expression!

Note that if a quadratic equation is presented to you in this format, taking the square root of both sides is by far the most convenient solution method.

9.2: Completing the square

The point of completing the square is to force any quadratic equation into the format in 9.1 above. This is done by simple algebra tricks, but the process can still be very confusing:

  1. Get all variables on one side, constants on the other.
  2. Get a coefficient of 1 on the quadratic term.
  3. Guess the binomial you square to get all the same variable parts.
  4. Compensate for the constant you’ve implicitly added to the variable side.

9.3: The Quadratic Formula.

If you complete the square to solve the general quadratic equation ax^2 + bx + c = 0 , then the solutions turn out to be:

(^2 ) 2

x b^ b^ ac a

=^ −^ ±^ −

Thus, all you have to do to solve any quadratic equation is to write it in the form ax^2 + bx + c = 0 , identify the constants a b , and c and plug into the formula carefully. The solutions should be simplified as far as possible – irrational solutions are normally written as a single fraction with a ± in the numerator, while rational solutions (integers or fractions) are written separately.

9.4: Complex numbers.

If we study an expression like − 4 , we realize that it can’t be a real number, since squaring a real number always yields a positive result. The problem is the factor of -1, so we gave it a name: i.

i = − 1 or i^2 = − 1

Using this definition, we were able to simplify square roots of negative numbers, but we also discovered that we could define a completely new type of number: complex numbers.

Complex numbers consist of a real part and an imaginary part. They can always be written in the form: a + bi

When performing operations with complex numbers, you should always write the answer in this form (this is standard math “grammar”).

Addition and subtraction of complex numbers happens by just combining like terms as usual (combine constants separately from terms containing i ).

Multiplication of complex numbers is just like multiplication of binomials except for one thing -- i^2 should be replaced by − 1.

Division was more complicated: in order to write the result in the correct form, we needed to force the denominator to be a real number. This was accomplished by multiplying the numerator and denominator by the conjugate of the denominator.