Algebra II : Discriminants

Study concepts, example questions & explanations for Algebra II

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Example Questions

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Example Question #296 : Algebra Ii

Given , what is the value of the discriminant?

Possible Answers:

Correct answer:

Explanation:

In general, the discriminant is .

In this particual case .

Plug in these three values and simplify:

Example Question #1 : Discriminants

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

Possible Answers:

-7, 2 real solutions

-7, 2 imaginary solutions

57, 2 imaginary solutions

57, 1 real solution

57, 2 real solutions

Correct answer:

57, 2 real solutions

Explanation:

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 #1 : Discriminants

The equation 

has two imaginary solutions. 

For what positive integer values of  is this possible?

Possible Answers:

All positive integers

Correct answer:

Explanation:

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 #3 : Discriminants

The equation 

has two real solutions. 

For what positive integer values of  is this possible?

Possible Answers:

All positive integers

Correct answer:

Explanation:

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 #2 : Discriminants

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

Possible Answers:

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

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.

Correct answer:

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

Explanation:

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 #3 : Discriminants

Find the discriminant, , in the following quadratic expression:

Possible Answers:

Correct answer:

Explanation:

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 #6 : Discriminants

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

How many solutions does the function  have?

Possible Answers:

1 real solution; 1 imaginary solution

No solution

2 imaginary solutions

2 real solutions

1 real solution

Correct answer:

2 real solutions

Explanation:

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 #7 : Discriminants

Determine the discriminant of the following quadratic equation .

Possible Answers:

Correct answer:

Explanation:

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

Example Question #4 : Discriminants

What is the discriminant for the function ?

Possible Answers:

Correct answer:

Explanation:

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 #9 : Discriminants

How many solutions does the quadratic  have?

Possible Answers:

 immaginary solutions

 real solution

no solutions

 real solution and  immaginary solution

 real solutions

Correct answer:

 real solutions

Explanation:

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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