Understand and apply concepts of equations - HiSet: High School Equivalency Test: Math

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.

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

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 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.

The first step to solving an equation where is in a radical is to isolate the radical. To do this, we need to subtract the 5 from both sides.

Now that the radical is isolated, clear the radical by raising both sides to the power of 3. Note:

Now we want to isolate the term. First, subtract the 5 from both sides.

Finally, divide both sides by to 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.

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 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.

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, 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, 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.

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.