Algebra II : Discriminants

Example Questions

Example Question #91 : Understanding Quadratic Equations

Given , what is the value of the discriminant?

Explanation:

In general, the discriminant is .

In this particual case .

Plug in these three values and simplify:

Example Question #1 : Discriminants

The equation

has two imaginary solutions.

For what positive integer values of  is this possible?

All positive integers

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 #2 : Discriminants

The equation

has two real solutions.

For what positive integer values of  is this possible?

All positive integers

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 #3 : Discriminants

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

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

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

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

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

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

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 #4 : Discriminants

Find the discriminant, , in the following quadratic expression:

Explanation:

.

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 #5 : Discriminants

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

How many solutions does the function  have?

No solution

2 real solutions

2 imaginary solutions

1 real solution; 1 imaginary solution

1 real solution

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 #6 : Discriminants

Determine the discriminant of the following quadratic equation .

Explanation:

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

Example Question #1 : Discriminants

What is the discriminant for the function ?

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 #1 : Discriminants

How many solutions does the quadratic  have?

real solution

no solutions

real solutions

real solution and  immaginary solution

immaginary solutions

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.

Example Question #9 : Discriminants

How many real roots are there to the following equation:

None of the above