8/20/2020 Sophia :: Welcome
Score 20/20
You passed this Milestone
20 questions were answered correctly.
UNIT 4 — MILESTONE 4
1
Select the quadratic equation that has no real solution.
RATIONALE
We can tell if a quadratic has no real solutions by using the quadratic formula. We
can use the discriminant, or the value underneath the square root. Because the
discriminant is underneath a square root sign, it must not have a negative value. If it
is greater than or equal to zero, it will have real solutions. If it is negative, it will have
non-real solutions.
This is the expression for the discriminant. For each quadratic equation, we can
substitute the appropriate values into this expression and determine if it will have real
or non-real solutions.
For this answer choice, plug the coefficients into the expression for the
discriminant, and determine if the discriminant is negative.
https://www.sophia.org/spcc/college-algebra-3/milestone_take_feedbacks/5542354 1/19
,8/20/2020 Sophia :: Welcome
In the equation , , , and . Now that
the appropriate values are plugged in, evaluate the discriminant.
squared is and times times is . Next, find the
difference between and .
minus is . Since this value is non-negative, the equation
has at least one real solution.
For this answer choice, plug the coefficients into the expression for the
discriminant, and determine if the discriminant is negative.
In the equation , , , and . Now that
the appropriate values are plugged in, evaluate the discriminant.
squared is and times times is . Next, find the
difference between and .
minus is . Since this value is non-negative, the equation
has at least one real solution.
For this answer choice, plug the coefficients into the expression for the
discriminant, and determine if the discriminant is negative.
In the equation , , , and . Now that
the appropriate values are plugged in, evaluate the discriminant.
squared is and times times is . Next, find the
difference between and .
minus is . Since this value is non-negative, the equation
has at least one real solution.
For this answer choice, plug the coefficients into the expression for the
discriminant, and determine if the discriminant is negative.
In the equation , , , and . Now that
the appropriate values are plugged in, evaluate the discriminant.
squared is and times times is . Next, find the difference
between and .
minus is . Since this value is negative, the equation
has no real solutions.
CONCEPT
https://www.sophia.org/spcc/college-algebra-3/milestone_take_feedbacks/5542354 2/19
, 8/20/2020 Sophia :: Welcome
Quadratic Equations with No Real Solution
2
Write the following expression as a single complex number.
RATIONALE
When dividing complex numbers, we must clear imaginary numbers
from the denominator. To do so, we can use the complex conjugate of the
denominator to create a second fraction. The complex conjugate of
is .
In this second fraction, the complex conjugate is the numerator
and the denominator. Next, multiply across the numerators and
denominators using FOIL.
In both the numerator and denominator, we multiplied the first, outside,
inside and last terms together. Now, we can evaluate each multiplication.
Once we have evaluated the multiplication, we can combine any like
terms.
In the numerator, we can combine and to get . In the
denominator, and cancel each other out. The next step is to
https://www.sophia.org/spcc/college-algebra-3/milestone_take_feedbacks/5542354 3/19
Score 20/20
You passed this Milestone
20 questions were answered correctly.
UNIT 4 — MILESTONE 4
1
Select the quadratic equation that has no real solution.
RATIONALE
We can tell if a quadratic has no real solutions by using the quadratic formula. We
can use the discriminant, or the value underneath the square root. Because the
discriminant is underneath a square root sign, it must not have a negative value. If it
is greater than or equal to zero, it will have real solutions. If it is negative, it will have
non-real solutions.
This is the expression for the discriminant. For each quadratic equation, we can
substitute the appropriate values into this expression and determine if it will have real
or non-real solutions.
For this answer choice, plug the coefficients into the expression for the
discriminant, and determine if the discriminant is negative.
https://www.sophia.org/spcc/college-algebra-3/milestone_take_feedbacks/5542354 1/19
,8/20/2020 Sophia :: Welcome
In the equation , , , and . Now that
the appropriate values are plugged in, evaluate the discriminant.
squared is and times times is . Next, find the
difference between and .
minus is . Since this value is non-negative, the equation
has at least one real solution.
For this answer choice, plug the coefficients into the expression for the
discriminant, and determine if the discriminant is negative.
In the equation , , , and . Now that
the appropriate values are plugged in, evaluate the discriminant.
squared is and times times is . Next, find the
difference between and .
minus is . Since this value is non-negative, the equation
has at least one real solution.
For this answer choice, plug the coefficients into the expression for the
discriminant, and determine if the discriminant is negative.
In the equation , , , and . Now that
the appropriate values are plugged in, evaluate the discriminant.
squared is and times times is . Next, find the
difference between and .
minus is . Since this value is non-negative, the equation
has at least one real solution.
For this answer choice, plug the coefficients into the expression for the
discriminant, and determine if the discriminant is negative.
In the equation , , , and . Now that
the appropriate values are plugged in, evaluate the discriminant.
squared is and times times is . Next, find the difference
between and .
minus is . Since this value is negative, the equation
has no real solutions.
CONCEPT
https://www.sophia.org/spcc/college-algebra-3/milestone_take_feedbacks/5542354 2/19
, 8/20/2020 Sophia :: Welcome
Quadratic Equations with No Real Solution
2
Write the following expression as a single complex number.
RATIONALE
When dividing complex numbers, we must clear imaginary numbers
from the denominator. To do so, we can use the complex conjugate of the
denominator to create a second fraction. The complex conjugate of
is .
In this second fraction, the complex conjugate is the numerator
and the denominator. Next, multiply across the numerators and
denominators using FOIL.
In both the numerator and denominator, we multiplied the first, outside,
inside and last terms together. Now, we can evaluate each multiplication.
Once we have evaluated the multiplication, we can combine any like
terms.
In the numerator, we can combine and to get . In the
denominator, and cancel each other out. The next step is to
https://www.sophia.org/spcc/college-algebra-3/milestone_take_feedbacks/5542354 3/19
