### All Algebra II Resources

## Example Questions

### Example Question #1 : Discriminants

Given , what is the value of the discriminant?

**Possible Answers:**

**Correct answer:**

In general, the discriminant is .

In this particual case .

Plug in these three values and simplify:

### Example Question #2 : Discriminants

Find the value of the determinant and state the number of real and imaginary solutions.

**Possible Answers:**

57, 2 real solutions

57, 2 imaginary solutions

-7, 2 imaginary solutions

-7, 2 real solutions

57, 1 real solution

**Correct answer:**

57, 2 real solutions

Given the quadratic equation of

The formula for the discriminant is (remember this as a part of the quadratic formula?)

Plugging in values to the discriminant equation:

So the determinant is 57. What does that mean for our solutions? Since it is a positive number, we know that we will have 2 real solutions. So the answer is:

57, 2 real solutions

### Example Question #3 : Discriminants

The equation

has two imaginary solutions.

For what positive integer values of is this possible?

**Possible Answers:**

All positive integers

**Correct answer:**

For the equation

to have two imaginary solutions, its discriminant must be negative. Set and solve for in the inequality

Therefore, if is a positive integer, it must be in the set .

### Example Question #4 : Discriminants

The equation

has two real solutions.

For what positive integer values of is this possible?

**Possible Answers:**

All positive integers

**Correct answer:**

For the equation

to have two real solutions, its discriminant must be positive. Set and solve for in the inequality

Therefore, if is a positive integer, it must be in the set

### Example Question #5 : Discriminants

What is the discriminant of the following quadratic equation? Are its roots real?

**Possible Answers:**

The equation's discriminant is and the its roots are not real.

The equation's discriminant is and its roots are not real.

The equation's roots are not real; therefore, it does not have a discriminant.

The equation's discriminant is and its roots are real.

The equation's discriminant is and its roots are real.

**Correct answer:**

The equation's discriminant is and the its roots are not real.

The "discriminant" is the name given to the expression that appears under the square root (radical) sign in the quadratic formula, where , , and are the numbers in the general form of a quadratic trinomial: . If the discriminant is positive, the equation has real roots, and if it is negative, we have imaginary roots. In this case, , , and , so the discriminant is , and because it is negative, this equation's roots are not real.

### Example Question #6 : Discriminants

Find the discriminant, , in the following quadratic expression:

**Possible Answers:**

**Correct answer:**

Remember the quadratic formula:

.

The discriminant in the quadratic formula is the term that appears under the square root symbol. It tells us about the nature of the roots.

So, to find the discriminant, all we need to do is compute for our equation, where .

We get .

### Example Question #7 : Discriminants

Choose the answer that is the most correct out of the following options.

How many solutions does the function have?

**Possible Answers:**

2 real solutions

1 real solution; 1 imaginary solution

No solution

1 real solution

2 imaginary solutions

**Correct answer:**

2 real solutions

The number of roots can be found by looking at the discriminant. The discriminant is determined by . For this function, ,, and . Therfore, . When the discriminant is positive, there are two real solutions to the function.

### Example Question #8 : Discriminants

Determine the discriminant of the following quadratic equation .

**Possible Answers:**

**Correct answer:**

The discriminant is found using the equation . So for the function , ,, and . Therefore the equation becomes .

### Example Question #9 : Discriminants

What is the discriminant for the function ?

**Possible Answers:**

**Correct answer:**

Given that quadratics can be written as . The discriminant can be found by looking at or the value under the radical of the quadratic formula. Using substiution and order of operations we can find this value of the discriminant of this quadratic equation.

### Example Question #10 : Discriminants

How many solutions does the quadratic have?

**Possible Answers:**

no solutions

real solutions

immaginary solutions

real solution

real solution and immaginary solution

**Correct answer:**

real solutions

The discrimiant will determine how many solutions a quadratic has. If the discriminant is positive, then there are two real solutions. If it is negative then there are two immaginary solutions. If it is equal to zero then there is one repeated solution.

Given that quadratics can be written as . The discriminant can be found by looking at or the value under the radical of the quadratic formula. Using substiution and order of operations we can find this value of the discriminant of this quadratic equation.

The discriminant is positive; therefore, there are two real solutions to this quadratic.

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