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You passed this Milestone
20 questions were answered correctly.
UNIT 4 — MILESTONE 4
1
Divide the following expression.
RATIONALE
Start by rewriting the expression into multiple fractions with as the denominator.
Remember to use the correct signs (addition or subtraction) between the fractions.
Now that we have individual fractions, we can simplify each fraction. To do this, cancel
out common factors in the numerator and denominator. Let's consider the first set, .
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simplifies to because we can factor out from both terms. Next, consider the
second set, .
simplifies to because we can factor out from both terms. Next, consider the
third set, .
simplifies to because we can factor out from both terms. The expression
can be simplified to .
CONCEPT
Polynomials Divided by Monomials
2
Consider the quadratic equation .
Find the solutions by factoring.
RATIONALE
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The expression, , is a basic quadratic equation because it is in the form
x squared plus b x plus c and can be factored as
left parenthesis x plus p right parenthesis left parenthesis x plus q right
parenthesis
. We must identify two integers, p and q, whose product is the constant term,
, and whose sum is the x-term coefficient, .
Two integers that multiply to and add to are and . Now that we have
identified p as and q as , we can substitute the values into the factored
quadratic expression of
left parenthesis x plus p right parenthesis left parenthesis x plus q right
parenthesis
.
If any factor of an algebraic expression equals zero, the entire expression has a value
of zero. We can set each factor equal to zero and solve the simpler equations to find
the solutions to the quadratic equation.
Setting each factor to zero will allow us to find the solutions. We can now solve for
x in each factor.
When we set the first factor equal to zero, the solution is . When we set
the second factor equal to zero, the solution is . The two solutions to
the equation are .
CONCEPT
Solving Quadratic Equations
3
Consider the quadratic inequality .
What is the solution set?
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RATIONALE
To solve a quadratic inequality, first rewrite it as
an equation set equal to zero.
Once we have a quadratic equation, we can solve
it by factoring. The expression is a
basic quadratic equation because it is in the form
x squared plus b x plus c and can be factored
as
left parenthesis x plus p right parenthesis
left parenthesis x plus q right parenthesis
. Next, identify two integers, p and q, whose
product is the constant term, , and whose sum
is the x-term coefficient, .
Two integers that multiply to and add to
are and . Now that we have identified p
and q, we can substitute the values into the
factored quadratic expression,
left parenthesis x plus p right parenthesis
left parenthesis x plus q right parenthesis
.
If any factor of an algebraic expression equals
zero, the entire expression has a value of zero.
Next, set each factor equal to zero and solve the
simpler equations to find the solutions to the
quadratic equation.
The solutions can be found by setting each factor
to zero. Next, solve for x in each factor.
Setting the first factor equal to zero, the
solution is . Setting the second factor
equal to zero, the solution is . These solutions
will create three intervals on the number line.
Next, choose any value that fits within each
interval. It doesn’t matter which values we
choose as test values, but we should make them
as simple as possible. Use them as x-values to be
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Score 20/20
You passed this Milestone
20 questions were answered correctly.
UNIT 4 — MILESTONE 4
1
Divide the following expression.
RATIONALE
Start by rewriting the expression into multiple fractions with as the denominator.
Remember to use the correct signs (addition or subtraction) between the fractions.
Now that we have individual fractions, we can simplify each fraction. To do this, cancel
out common factors in the numerator and denominator. Let's consider the first set, .
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,2/5/2021 Sophia :: Welcome
simplifies to because we can factor out from both terms. Next, consider the
second set, .
simplifies to because we can factor out from both terms. Next, consider the
third set, .
simplifies to because we can factor out from both terms. The expression
can be simplified to .
CONCEPT
Polynomials Divided by Monomials
2
Consider the quadratic equation .
Find the solutions by factoring.
RATIONALE
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,2/5/2021 Sophia :: Welcome
The expression, , is a basic quadratic equation because it is in the form
x squared plus b x plus c and can be factored as
left parenthesis x plus p right parenthesis left parenthesis x plus q right
parenthesis
. We must identify two integers, p and q, whose product is the constant term,
, and whose sum is the x-term coefficient, .
Two integers that multiply to and add to are and . Now that we have
identified p as and q as , we can substitute the values into the factored
quadratic expression of
left parenthesis x plus p right parenthesis left parenthesis x plus q right
parenthesis
.
If any factor of an algebraic expression equals zero, the entire expression has a value
of zero. We can set each factor equal to zero and solve the simpler equations to find
the solutions to the quadratic equation.
Setting each factor to zero will allow us to find the solutions. We can now solve for
x in each factor.
When we set the first factor equal to zero, the solution is . When we set
the second factor equal to zero, the solution is . The two solutions to
the equation are .
CONCEPT
Solving Quadratic Equations
3
Consider the quadratic inequality .
What is the solution set?
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RATIONALE
To solve a quadratic inequality, first rewrite it as
an equation set equal to zero.
Once we have a quadratic equation, we can solve
it by factoring. The expression is a
basic quadratic equation because it is in the form
x squared plus b x plus c and can be factored
as
left parenthesis x plus p right parenthesis
left parenthesis x plus q right parenthesis
. Next, identify two integers, p and q, whose
product is the constant term, , and whose sum
is the x-term coefficient, .
Two integers that multiply to and add to
are and . Now that we have identified p
and q, we can substitute the values into the
factored quadratic expression,
left parenthesis x plus p right parenthesis
left parenthesis x plus q right parenthesis
.
If any factor of an algebraic expression equals
zero, the entire expression has a value of zero.
Next, set each factor equal to zero and solve the
simpler equations to find the solutions to the
quadratic equation.
The solutions can be found by setting each factor
to zero. Next, solve for x in each factor.
Setting the first factor equal to zero, the
solution is . Setting the second factor
equal to zero, the solution is . These solutions
will create three intervals on the number line.
Next, choose any value that fits within each
interval. It doesn’t matter which values we
choose as test values, but we should make them
as simple as possible. Use them as x-values to be
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