Download Quadratic equations and factoring

Solving quadratic equations by factoring
Quadratic Equation
Standard form:
,
Where a does not equal 0.
Example 1
Solve
by factoring.
Step 1: Simplify each side if needed.
This quadratic equation is already simplified.
Step 2: Write in standard form,
, if needed.
This quadratic equation is already in standard form.
Step 3: Factor.
Step 4: Use the Zero-Product Principle
AND
Step 5: Solve for the linear equation(s) set up in step 4.
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*Use Zero-Product Principle
*Solve the first linear equation
*Solve the second linear equation
There are two solutions to this quadratic equation:
x = -5 and x = 2.
Example 2:
Solve
by factoring.
Step 1: Simplify each side if needed.
*Mult. both sides by LCD of 6 to clear
fractions
Step 2: Write in standard form,
, if needed.
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*Inverse of add. 16 is sub. 16
*Quad. eq. in standard form
Step 3: Factor.
*Quad. eq. in standard form
*Factor the diff. of two squares
Step 4: Use the Zero-Product Principle
AND
Step 5: Solve for the linear equation(s) set up in step 4.
*Use Zero-Product Principle
*Solve the first linear equation
*Solve the second linear equation
There are two solutions to this quadratic equation: x = -4/5 and x =
4/5.
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Example 3:
Solve
by factoring.
Step 1: Simplify each side if needed.
*Use Dist. Prop. to clear the ( )
Step 2: Write in standard form,
, if needed.
*Inverse of add. 2 is sub. 2
*Quad. eq. in standard form
Step 3: Factor.
*Quad. eq. in standard form
*Factor the trinomial
Step 4: Use the Zero-Product Principle
AND
Step 5: Solve for the linear equation(s) set up in step 4.
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*Use Zero-Product Principle
*Solve the first linear equation
*Solve the second linear equation
There are two solutions to this quadratic equation:
x = -2/3 and x = 1/2.
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Solving Quadratic Equations
by the Square Root Method
You can solve a quadratic equation by the square root method if
you can write it in the form
.
Step 1: Write the quadratic equation in the form
if needed.
A and B represent algebraic expressions. When you have the
quadratic equation written in this form, it allows you to use the
square root method described in step 2.
If it is not in this form,
, move any term(s) to the
appropriate side by using the addition/subtraction or
multiplication/division property of equality.
Step 2: Apply the square root method.
If A and B are algebraic expressions such that
,
then
also written
,
.
In other words, if you have an expression squared set equal to
another expression, the inverse operation to solve it is to take
the square root of both sides. Since both a positive and its
opposite squared result in the same answer, then you will have
two answers, plus or minus the square root of B.
Step 3: Solve for the linear equation(s) set up in step 2.
After applying the square root method to a quadratic equation
you will end up with either one or two linear equations to solve.
Most times you will have two linear equations, but if B is equal
to 0, then you will only have one since plus or minus 0 is only
one number.
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Example 4: Solve
by using the square root method.
Step 1: Write the quadratic equation in the form
if needed
AND
Step 2: Apply the square root method.
*Written in the form
*Apply the sq. root method
*There are 2 solutions
Step 3: Solve for the linear equation(s) set up in step 2.
*Sq. root of 16 = 4
*Neg. sq. root of 16 = - 4
There are two solutions to this quadratic equation: x = 4 and x = 4.
Example 5: Solve
by using the square root method.
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Step 1: Write the quadratic equation in the form
if needed
AND
Step 2: Apply the square root method.
Note how this quadratic equation is not in the form
to begin
with. The 5 is NOT part of the expression being squared on the left
by
side of the equation. We can easily write it in the form
dividing both sides by 5.
*Not in the form
*Inv. of mult. by 5 is div. by 5
*Written in the form
*Apply the sq. root method
*There are 2 solutions
Step 3: Solve for the linear equation(s) set up in step 2.
*Sq. root of 4 = 2
*Neg. sq. root of 4 = -2
There are two solutions to this quadratic equation: x = 2 and x = 2.
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Example 6: Solve
by using the square root method.
Step 1: Write the quadratic equation in the form
if needed
AND
Step 2: Apply the square root method.
*Written in the
form
*Apply the sq. root method
*There are 2 solutions
Step 3: Solve for the linear equation(s) set up in step 2.
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*Sq. root of 20 = 2 sq. root of 5
*Solve for x
*Neg. sq. root of 20 = -2 sq. root of 5
*Solve for x
There are two solutions to this quadratic equation: x =
and x =
.
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Solving Quadratic Equations
by Completing the Square
You can solve ANY quadratic equation by completing the square.
This comes in handy when a quadratic equation does not factor or
is difficult to factor.
Step 1:Make sure that the coefficient on the
term is equal to 1.
If the coefficient of the
term is already 1, then proceed to step 2.
If the coefficient of the
by that coefficient.
term is not equal to 1, then divide both sides
Step 2:Isolate the
and x terms.
In other words, rewrite it so that the
the constant is on the other side.
and x terms are on one side and
Step 3: Complete the square.
At this point we will be creating a perfect square trinomial (PST).
Recall that a PST is a trinomial of the form
and it factors
. When it is in that form it will allow us to
in the form
continue onto the next step and take the square root of both sides and
find a solution.
We need to find a number that we can add to the
that we have a PST.
and x terms so
We can get that magic number by doing the following:
If we have
we can complete it’s square by adding the
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constant
In other words, we complete the square by taking ½ of b (the
coefficient of the x term) and then squaring it. Make sure you
remember to add it to BOTH sides to keep the equation balanced.
Step 4: Factor the perfect square trinomial (created in step 3) as a
binomial squared.
If you need a review on factoring a perfect square trinomial, feel free to
go to Tutorial 7: Factoring Polynomials.
Step 5: Solve the equation in step 4 by using the square root
method.
Example 7: Solve
by completing the square.
Step 1:Make sure that the coefficient on the
The coefficient of the
Step 2: Isolate the
The
term is equal to 1.
term is already 1.
and x terms.
and x terms are already isolated.
Step 3: Complete the square.
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*b is the coefficient of the x term
*Complete the square by taking 1/2 of b and squaring it
*Add constant found above to BOTH sides
of the eq.
*This creates a PST on the left side of eq.
Step 4: Factor the perfect square trinomial (created in step 3) as a
binomial squared.
*Factor the PST
Step 5: Solve the equation in step 4 by using the square root
method.
*Written in the form
*Apply the sq. root method
*There are 2 solutions
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There are two solutions to this quadratic equation: x = 9 and x =
1.
Example 8: Solve
by completing the square.
Step 1:Make sure that the coefficient on the
term is equal to 1.
term is not 1 to begin with. We can
Note how the coefficient on the
easily fix that by dividing both sides by that coefficient, which in this
case is 3 .
*Divide both sides by 3
*Coefficient of
Step 2: Isolate the
term is now 1
and x terms.
and x terms are not isolated to begin with. We can
Note how the
easily fix that by moving the constant to the other side of the equation.
*Inverse of add. 3 is sub. 3
*
and x terms are now isolated
Step 3: Complete the square.
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*b is the coefficient of the x term
*Complete the square by taking 1/2 of b and
squaring it
*Add constant found above to BOTH sides
of the eq.
*This creates a PST on the left side of eq.
Step 4: Factor the perfect square trinomial (created in step 3) as a
binomial squared.
*Factor the PST
Step 5: Solve the equation in step 4 by using the square root
method.
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*Written in the form
*Apply the sq. root method
*There are 2 solutions
*Square root of a negative 1 is i
*Square root of a negative 1 is i
There are two solutions to this quadratic equation: x =
and x =
.
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