Quadratic equations - HiSET
Card 0 of 44
What is the vertex of the following quadratic polynomial?
What is the vertex of the following quadratic polynomial?
Tap to see back →
Given a quadratic function
the vertex will always be
.
Thus, since our function is
, 
, and 
.
We plug these variables into the formula to get the vertex as
.
Hence, the vertex of
is
.
Given a quadratic function
the vertex will always be
Thus, since our function is
We plug these variables into the formula to get the vertex as
Hence, the vertex of
is
Which of the following expressions represents the discriminant of the following polynomial?
Which of the following expressions represents the discriminant of the following polynomial?
Tap to see back →
The discriminant of a quadratic polynomial
is given by
.
Thus, since our quadratic polynomial is
,
, 
, and 
.
Plugging these values into the discriminant equation, we find that the discriminant is
.
The discriminant of a quadratic polynomial
is given by
Thus, since our quadratic polynomial is
Plugging these values into the discriminant equation, we find that the discriminant is
Give the nature of the solution set of the equation
Give the nature of the solution set of the equation
Tap to see back →
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
This can be done by adding 17 to both sides:
The key to determining the nature of the solution set is to examine the discriminant
. Setting 
, the value of the discriminant is
This value is negative. Consequently, the solution set comprises two imaginary numbers.
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
This can be done by adding 17 to both sides:
The key to determining the nature of the solution set is to examine the discriminant
This value is negative. Consequently, the solution set comprises two imaginary numbers.
Which of the following polynomial equations has exactly one solution?
Which of the following polynomial equations has exactly one solution?
Tap to see back →
A polynomial equation of the form
has one and only one (real) solution if and only if its discriminant is equal to zero - that is, if its coefficients satisfy the equation
In each of the choices, 
and 
, so it suffices to determine the value of 
which satisfies this equation. Substituting, we get
Solve for 
by first adding 400 to both sides:
Take the square root of both sides:
The choice that matches this value of 
is the equation
A polynomial equation of the form
has one and only one (real) solution if and only if its discriminant is equal to zero - that is, if its coefficients satisfy the equation
In each of the choices, 
Solve for 
Take the square root of both sides:
The choice that matches this value of 
Give the nature of the solution set of the equation
Give the nature of the solution set of the equation
Tap to see back →
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
To accomplish this, first, multiply the binomials on the left using the FOIL technique:
Collect like terms:
Now, subtract 18 from both sides:
The key to determining the nature of the solution set is to examine the discriminant 
. Setting 
, the value of the discriminant is
The discriminant is a positive number, so there are two real solutions. Since 73 is not a perfect square, the solutions are irrational.
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
To accomplish this, first, multiply the binomials on the left using the FOIL technique:
Collect like terms:
Now, subtract 18 from both sides:
The key to determining the nature of the solution set is to examine the discriminant 
The discriminant is a positive number, so there are two real solutions. Since 73 is not a perfect square, the solutions are irrational.
Give the nature of the solution set of the equation
Give the nature of the solution set of the equation
Tap to see back →
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
To accomplish this, first, multiply the binomials on the left using the FOIL technique:
Collect like terms:
Add 6 to both sides:
The key to determining the nature of the solution set is to examine the discriminant . Setting 
, the value of the discriminant is
The discriminant is a positive number; furthermore, it is a perfect square, being equal to the square of 11. Therefore, the solution set comprises two rational solutions.
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
To accomplish this, first, multiply the binomials on the left using the FOIL technique:
Collect like terms:
Add 6 to both sides:
The key to determining the nature of the solution set is to examine the discriminant 
The discriminant is a positive number; furthermore, it is a perfect square, being equal to the square of 11. Therefore, the solution set comprises two rational solutions.
Give the nature of the solution set of the equation
.
Give the nature of the solution set of the equation
Tap to see back →
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
This can be done by simply switching the first and second terms:
The key to determining the nature of the solution set is to examine the discriminant 
. Setting 
, the value of the discriminant is
The discriminant has a negative value. It follows that the solution set comprises two imaginary values.
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
This can be done by simply switching the first and second terms:
The key to determining the nature of the solution set is to examine the discriminant 
The discriminant has a negative value. It follows that the solution set comprises two imaginary values.
Which of the following polynomial equations has exactly one solution?
Which of the following polynomial equations has exactly one solution?
Tap to see back →
A polynomial equation of the standard form
has one and only one (real) solution if and only if its discriminant is equal to zero - that is, if its coefficients satisfy the equation
Each of the choices can be rewritten in standard form by subtracting the term on the right from both sides. One of the choices can be rewritten as follows:
By similar reasoning, the other four choices can be written:
In each of the five standard forms, 
and 
, so it is necessary to determine the value of 
that produces a zero discriminant. Substituting accordingly:
Add 900 to both sides and take the square root:
Of the five standard forms,
fits this condition. This is the standard form of the equation
,
the correct choice.
A polynomial equation of the standard form
has one and only one (real) solution if and only if its discriminant is equal to zero - that is, if its coefficients satisfy the equation
Each of the choices can be rewritten in standard form by subtracting the term on the right from both sides. One of the choices can be rewritten as follows:
By similar reasoning, the other four choices can be written:
In each of the five standard forms, 
Add 900 to both sides and take the square root:
Of the five standard forms,
fits this condition. This is the standard form of the equation
the correct choice.
Give the nature of the solution set of the equation
Give the nature of the solution set of the equation
Tap to see back →
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
To accomplish this, first, multiply the binomials on the left using the FOIL technique:
Collect like terms:
Now, add 18 to both sides:
The key to determining the nature of the solution set is to examine the discriminant . Setting 
, the value of the discriminant is
This discriminant is negative. Consequently, the solution set comprises two imaginary numbers.
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
To accomplish this, first, multiply the binomials on the left using the FOIL technique:
Collect like terms:
Now, add 18 to both sides:
The key to determining the nature of the solution set is to examine the discriminant 
This discriminant is negative. Consequently, the solution set comprises two imaginary numbers.
Give the nature of the solution set of the equation
Give the nature of the solution set of the equation
Tap to see back →
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
This can be done by switching the first and third terms on the left:
The key to determining the nature of the solution set is to examine the discriminant
. Setting 
, the value of the discriminant is
.
The discriminant is a positive number but not a perfect square. Therefore, there are two irrational solutions.
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
This can be done by switching the first and third terms on the left:
The key to determining the nature of the solution set is to examine the discriminant
The discriminant is a positive number but not a perfect square. Therefore, there are two irrational solutions.
Give the nature of the solution set of the equation
Give the nature of the solution set of the equation
Tap to see back →
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
This can be done by subtracting 34 from to both sides:
The key to determining the nature of the solution set is to examine the discriminant
. Setting 
, the value of the discriminant is
.
The discriminant is a positive number but not a perfect square. Therefore, there are two irrational solutions.
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
This can be done by subtracting 34 from to both sides:
The key to determining the nature of the solution set is to examine the discriminant
The discriminant is a positive number but not a perfect square. Therefore, there are two irrational solutions.
Give the nature of the solution set of the equation
Give the nature of the solution set of the equation
Tap to see back →
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
To accomplish this, first, multiply the binomials on the left using the FOIL technique:
Collect like terms:
Now, add 18 to both sides:
The key to determining the nature of the solution set is to examine the discriminant . Setting , the value of the discriminant is
This discriminant is negative. Consequently, the solution set comprises two imaginary numbers.
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
To accomplish this, first, multiply the binomials on the left using the FOIL technique:
Collect like terms:
Now, add 18 to both sides:
The key to determining the nature of the solution set is to examine the discriminant
This discriminant is negative. Consequently, the solution set comprises two imaginary numbers.
What is the vertex of the following quadratic polynomial?
What is the vertex of the following quadratic polynomial?
Tap to see back →
Given a quadratic function
the vertex will always be
.
Thus, since our function is
, , and .
We plug these variables into the formula to get the vertex as
.
Hence, the vertex of
is
.
Given a quadratic function
the vertex will always be
Thus, since our function is
We plug these variables into the formula to get the vertex as
Hence, the vertex of
is
Which of the following expressions represents the discriminant of the following polynomial?
Which of the following expressions represents the discriminant of the following polynomial?
Tap to see back →
The discriminant of a quadratic polynomial
is given by
.
Thus, since our quadratic polynomial is
,
, , and .
Plugging these values into the discriminant equation, we find that the discriminant is
.
The discriminant of a quadratic polynomial
is given by
Thus, since our quadratic polynomial is
Plugging these values into the discriminant equation, we find that the discriminant is
Give the nature of the solution set of the equation
Give the nature of the solution set of the equation
Tap to see back →
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
This can be done by adding 17 to both sides:
The key to determining the nature of the solution set is to examine the discriminant
. Setting , the value of the discriminant is
This value is negative. Consequently, the solution set comprises two imaginary numbers.
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
This can be done by adding 17 to both sides:
The key to determining the nature of the solution set is to examine the discriminant
This value is negative. Consequently, the solution set comprises two imaginary numbers.
Which of the following polynomial equations has exactly one solution?
Which of the following polynomial equations has exactly one solution?
Tap to see back →
A polynomial equation of the form
has one and only one (real) solution if and only if its discriminant is equal to zero - that is, if its coefficients satisfy the equation
In each of the choices, and , so it suffices to determine the value of which satisfies this equation. Substituting, we get
Solve for by first adding 400 to both sides:
Take the square root of both sides:
The choice that matches this value of is the equation
A polynomial equation of the form
has one and only one (real) solution if and only if its discriminant is equal to zero - that is, if its coefficients satisfy the equation
In each of the choices,
Solve for
Take the square root of both sides:
The choice that matches this value of
Give the nature of the solution set of the equation
Give the nature of the solution set of the equation
Tap to see back →
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
To accomplish this, first, multiply the binomials on the left using the FOIL technique:
Collect like terms:
Now, subtract 18 from both sides:
The key to determining the nature of the solution set is to examine the discriminant . Setting , the value of the discriminant is
The discriminant is a positive number, so there are two real solutions. Since 73 is not a perfect square, the solutions are irrational.
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
To accomplish this, first, multiply the binomials on the left using the FOIL technique:
Collect like terms:
Now, subtract 18 from both sides:
The key to determining the nature of the solution set is to examine the discriminant
The discriminant is a positive number, so there are two real solutions. Since 73 is not a perfect square, the solutions are irrational.
Give the nature of the solution set of the equation
Give the nature of the solution set of the equation
Tap to see back →
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
To accomplish this, first, multiply the binomials on the left using the FOIL technique:
Collect like terms:
Add 6 to both sides:
The key to determining the nature of the solution set is to examine the discriminant . Setting , the value of the discriminant is
The discriminant is a positive number; furthermore, it is a perfect square, being equal to the square of 11. Therefore, the solution set comprises two rational solutions.
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
To accomplish this, first, multiply the binomials on the left using the FOIL technique:
Collect like terms:
Add 6 to both sides:
The key to determining the nature of the solution set is to examine the discriminant
The discriminant is a positive number; furthermore, it is a perfect square, being equal to the square of 11. Therefore, the solution set comprises two rational solutions.
Give the nature of the solution set of the equation
.
Give the nature of the solution set of the equation
Tap to see back →
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
This can be done by simply switching the first and second terms:
The key to determining the nature of the solution set is to examine the discriminant . Setting , the value of the discriminant is
The discriminant has a negative value. It follows that the solution set comprises two imaginary values.
To determine the nature of the solution set of a quadratic equation, it is necessary to first express it in standard form
This can be done by simply switching the first and second terms:
The key to determining the nature of the solution set is to examine the discriminant
The discriminant has a negative value. It follows that the solution set comprises two imaginary values.
Which of the following polynomial equations has exactly one solution?
Which of the following polynomial equations has exactly one solution?
Tap to see back →
A polynomial equation of the standard form
has one and only one (real) solution if and only if its discriminant is equal to zero - that is, if its coefficients satisfy the equation
Each of the choices can be rewritten in standard form by subtracting the term on the right from both sides. One of the choices can be rewritten as follows:
By similar reasoning, the other four choices can be written:
In each of the five standard forms, and , so it is necessary to determine the value of that produces a zero discriminant. Substituting accordingly:
Add 900 to both sides and take the square root:
Of the five standard forms,
fits this condition. This is the standard form of the equation
,
the correct choice.
A polynomial equation of the standard form
has one and only one (real) solution if and only if its discriminant is equal to zero - that is, if its coefficients satisfy the equation
Each of the choices can be rewritten in standard form by subtracting the term on the right from both sides. One of the choices can be rewritten as follows:
By similar reasoning, the other four choices can be written:
In each of the five standard forms,
Add 900 to both sides and take the square root:
Of the five standard forms,
fits this condition. This is the standard form of the equation
the correct choice.