# HiSET: Math : Quadratic equations

## Example Questions

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### Example Question #1 : Quadratic Equations

What is the vertex of the following quadratic polynomial?

Explanation:

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

.

### Example Question #2 : Quadratic Equations

Which of the following expressions represents the discriminant of the following polynomial?

Explanation:

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

.

### Example Question #3 : Quadratic Equations

Which of the following polynomial equations has exactly one solution?

Explanation:

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

### Example Question #4 : Quadratic Equations

Give the nature of the solution set of the equation

Two rational solutions

One rational solution

Two irrational solutions

Two imaginary solutions

One imaginary solution

Two rational solutions

Explanation:

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:

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.

### Example Question #5 : Quadratic Equations

Which of the following polynomial equations has exactly one solution?

Explanation:

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.

### Example Question #6 : Quadratic Equations

Give the nature of the solution set of the equation

.

One imaginary solution

One rational solution

Two imaginary solutions

Two rational solutions

Two irrational solutions

Two imaginary solutions

Explanation:

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.

### Example Question #7 : Quadratic Equations

Give the nature of the solution set of the equation

One imaginary solution

Two rational solutions

One rational solution

Two imaginary solutions

Two irrational solutions

Two imaginary solutions

Explanation:

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.

### Example Question #8 : Quadratic Equations

Give the nature of the solution set of the equation

One rational solution

Two imaginary solutions

Two rational solutions

Two irrational solutions

One imaginary solution

Two irrational solutions

Explanation:

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.

### Example Question #9 : Quadratic Equations

Give the nature of the solution set of the equation

Two irrational solutions

Two imaginary solutions

One rational solution

One imaginary solution

Two rational solutions

Two imaginary solutions

Explanation:

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.

### Example Question #10 : Quadratic Equations

Give the nature of the solution set of the equation

One rational solution

Two rational solutions

Two irrational solutions

One imaginary solution

Two imaginary solutions

Two irrational solutions

Explanation:

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.

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