Algebraic Concepts - HiSET

The domain of the composition of functions is the set of all values that fall in the domain of such that falls in the domain of .

, a square root function, so the domain of is the set of all values that make the radicand a nonnegative number. Since the radicand is itself, the domain of is simply the set of all nonnegative numbers, or .

, a rational function, so its domain is the set of all numbers except those that make its denominator zero. if and only if , so from , we exclude only the value so that

;

that is,

Excluding this value from the domain, this leaves as the domain of .

Arrange and combine like terms (those with the same variable) as follows:

Since each term now has a different exponent for the variable, no further combining is possible. The simplified form has four terms.

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.

Generally speaking, in an equation a coefficient is a constant by which a variable is multiplied. For example, and are coefficients in the following equation:

In our equation, the following numbers are coefficients:

Step 1: Rearrange the terms from highest power to lowest power.

We will get: .

Step 2: We count the second term from starting from the left since it is the second highest term in the rearranged expression.

Step 3: Isolate the term.

The second term is

Step 4: Find the coefficient. The coefficient of a term is considered as the number before any variables. In this case, the coefficient is .

So, the answer is .

In an equation, a term is a single number or a variable. in our equation we have the following terms:

Step 1: We need to separate and arrange the terms in the jumbled expression given to us in the question..

We will separate the terms by the exponent value.

For , there is only one term:

For , there are no terms.

For , there is two terms:

For , there are two terms:

For , there are two terms:

For , there is only one term:

For , there are two terms:

For , there are three terms:

There are four constant terms:

Step 2: We will now add up the coefficients in each designation of terms. This will give us the answer.

Step 1: Add the terms based on their similarities...

and becomes .

and becomes .

and becomes

Step 2: Combine all the terms after "becomes" in step 1...

We add and get: .

Step 1: Subtract these terms by separating by exponents...

Step 2: Add all the simplified terms together...

Step 1: Find the Least Common Denominator of these fraction. We will list out multiples of each denominator until we find a common number for all three fractions...

The smallest common denominator is .

Step 2: Since the denominator is , we will convert all denominators to .

Step 3: Add up all the values of x...

Step 4: The result from step is the numerator and is the denominator. We will put these together.

Final Answer:

A polynomial function with zeroes 3 and 7 has as its factors and . The function is given to be quadratic, so this function is

.

Apply the FOIL method to rewrite the polynomial:

Collect like terms:

,

the correct choice.

Solve for in the equation

by isolating on the left side. Do this by reversing the operations in the reverse of the order of operations.

First, subtract 17 from both sides:

Now, divide both sides by 2:

One way to find 25% of this value is to multiply 41 by 25 and divide by 100:

,

the correct choice.

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

.

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

.

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.

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

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:

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

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.

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.