Understand and apply concepts of equations - HiSET
Card 0 of 84
Solve.

Solve.
In order to solve for the variable,
, we need to isolate it on the left side of the equation. We will do this by reversing the operations done to the variable by performing the opposite of each operation on both sides of the equation.
Let's begin by rewriting the given equation.

Subtract
from both sides of the equation.

Simplify.

Multiply both sides of the equation by
.

Solve.

In order to solve for the variable, , we need to isolate it on the left side of the equation. We will do this by reversing the operations done to the variable by performing the opposite of each operation on both sides of the equation.
Let's begin by rewriting the given equation.
Subtract from both sides of the equation.
Simplify.
Multiply both sides of the equation by .
Solve.
Compare your answer with the correct one above
Give the solution set of the inequality

Give the solution set of the inequality
can be rewritten as the compound inequality
.
The solution set will be the union of the two individual solution sets. Find the solution set of the first inequality as follows:
Isolate
by first subtracting
from both sides:



Divide both sides by
, reversing the direction of the inequality symbol, since you are dividing by a negative number:

.
In interval notation, this is the set
.
Find the solution set of the other inequality similarly:





In interval notation, this is the set
.
The union of these sets is the solution set:
.
can be rewritten as the compound inequality
.
The solution set will be the union of the two individual solution sets. Find the solution set of the first inequality as follows:
Isolate by first subtracting
from both sides:
Divide both sides by , reversing the direction of the inequality symbol, since you are dividing by a negative number:
.
In interval notation, this is the set .
Find the solution set of the other inequality similarly:
In interval notation, this is the set .
The union of these sets is the solution set: .
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What is the vertex of the following quadratic polynomial?

What is the vertex of the following quadratic polynomial?
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
,
, and
.
We plug these variables into the formula to get the vertex as
.
Hence, the vertex of
is
.
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Which of the following expressions represents the discriminant of the following polynomial?

Which of the following expressions represents the discriminant of the following polynomial?
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
,
,
, and
.
Plugging these values into the discriminant equation, we find that the discriminant is
.
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Give the nature of the solution set of the equation

Give the nature of the solution set of the equation
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
. Setting
, the value of the discriminant is
This value is negative. Consequently, the solution set comprises two imaginary numbers.
Compare your answer with the correct one above
Which of the following polynomial equations has exactly one solution?
Which of the following polynomial equations has exactly one solution?
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, 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
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Give the nature of the solution set of the equation

Give the nature of the solution set of the equation
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 . 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.
Compare your answer with the correct one above
Give the nature of the solution set of the equation

Give the nature of the solution set of the equation
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 . 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.
Compare your answer with the correct one above
Give the nature of the solution set of the equation
.
Give the nature of the solution set of the equation
.
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 . Setting
, the value of the discriminant is
The discriminant has a negative value. It follows that the solution set comprises two imaginary values.
Compare your answer with the correct one above
Which of the following polynomial equations has exactly one solution?
Which of the following polynomial equations has exactly one solution?
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, 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.
Compare your answer with the correct one above
Give the nature of the solution set of the equation

Give the nature of the solution set of the equation
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 . Setting
, the value of the discriminant is
This discriminant is negative. Consequently, the solution set comprises two imaginary numbers.
Compare your answer with the correct one above
Give the nature of the solution set of the equation

Give the nature of the solution set of the equation
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
. Setting
, the value of the discriminant is
.
The discriminant is a positive number but not a perfect square. Therefore, there are two irrational solutions.
Compare your answer with the correct one above
Give the nature of the solution set of the equation

Give the nature of the solution set of the equation
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
. Setting
, the value of the discriminant is
.
The discriminant is a positive number but not a perfect square. Therefore, there are two irrational solutions.
Compare your answer with the correct one above
Solve for
:

Solve for :
To solve for
in a literal equation, use the properties of algebra to isolate
on one side, just as if you were solving a regular equation.

Multiply both sides by
:


Subtract
from both sides:


Multiply both sides by
, distributing on the right:


,
the correct response.
To solve for in a literal equation, use the properties of algebra to isolate
on one side, just as if you were solving a regular equation.
Multiply both sides by :
Subtract from both sides:
Multiply both sides by , distributing on the right:
,
the correct response.
Compare your answer with the correct one above
Solve for
:

Assume
is positive.
Solve for :
Assume is positive.
To solve for
in a literal equation, use the properties of algebra to isolate
on one side, just as if you were solving a regular equation.

Subtract
from both sides:


Divide both sides by 9:


Take the square root of both sides:

Simplify the expression on the right by splitting it, and taking the square root of numerator and denominator:

,
the correct response.
To solve for in a literal equation, use the properties of algebra to isolate
on one side, just as if you were solving a regular equation.
Subtract from both sides:
Divide both sides by 9:
Take the square root of both sides:
Simplify the expression on the right by splitting it, and taking the square root of numerator and denominator:
,
the correct response.
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Solve for
:

Solve for :
To solve for
in a literal equation, use the properties of algebra to isolate
on one side, just as if you were solving a regular equation.

Subtract 20 from both sides:


Divide both sides by
:

,
the correct response.
To solve for in a literal equation, use the properties of algebra to isolate
on one side, just as if you were solving a regular equation.
Subtract 20 from both sides:
Divide both sides by :
,
the correct response.
Compare your answer with the correct one above
Solve for
:

Solve for :
To solve for
in a literal equation, use the properties of algebra to isolate
on one side, just as if you were solving a regular equation.
First, square both sides to eliminate the radical symbol:



Rewrite the expression on the right using the square of a binomial pattern:


Subtract 1 from both sides:

,
the correct response.
To solve for in a literal equation, use the properties of algebra to isolate
on one side, just as if you were solving a regular equation.
First, square both sides to eliminate the radical symbol:
Rewrite the expression on the right using the square of a binomial pattern:
Subtract 1 from both sides:
,
the correct response.
Compare your answer with the correct one above
Solve for
:

You my assume
is positive.
Solve for :
You my assume is positive.
To solve for
in a literal equation, use the properties of algebra to isolate
on one side, just as if you were solving a regular equation.
First, add
to both sides:



Take the positive square root of both sides:
,
the correct response.
To solve for in a literal equation, use the properties of algebra to isolate
on one side, just as if you were solving a regular equation.
First, add to both sides:
Take the positive square root of both sides:
,
the correct response.
Compare your answer with the correct one above
Solve for
:

Solve for :
To solve for
in a literal equation, use the properties of algebra to isolate
on one side, just as if you were solving a regular equation.
First, take the reciprocal of both sides:


Multiply both sides by
:


Distribute on the right:


Subtract 1 from both sides, rewriting 1 as
to facilitate subtraction:


,
the correct response.
To solve for in a literal equation, use the properties of algebra to isolate
on one side, just as if you were solving a regular equation.
First, take the reciprocal of both sides:
Multiply both sides by :
Distribute on the right:
Subtract 1 from both sides, rewriting 1 as to facilitate subtraction:
,
the correct response.
Compare your answer with the correct one above
Solve.

Solve.
In order to solve for the variable,
, we need to isolate it on the left side of the equation. We will do this by reversing the operations done to the variable by performing the opposite of each operation on both sides of the equation.
Let's begin by rewriting the given equation.

Subtract
from both sides of the equation.

Simplify.

Multiply both sides of the equation by
.

Solve.

In order to solve for the variable, , we need to isolate it on the left side of the equation. We will do this by reversing the operations done to the variable by performing the opposite of each operation on both sides of the equation.
Let's begin by rewriting the given equation.
Subtract from both sides of the equation.
Simplify.
Multiply both sides of the equation by .
Solve.
Compare your answer with the correct one above