Algebra II : Intermediate Single-Variable Algebra

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #41 : Discriminants

Determine the discriminant:  

Possible Answers:

Correct answer:

Explanation:

Write the formula for the discriminant.  The discriminant is the term inside the square root portion of the quadratic formula.

Substitute the known coefficients of the polynomial in standard form:

The answer is:  

Example Question #1440 : Algebra Ii

For the given polynomial function , find the relationship between  and  so that the function will have:

 

a) Two non-zero real roots 

b) One non-zero real root  

c) Two Complex Roots

 

 

Where the constants  and  can be any non-zero, positive real number. 

Possible Answers:

a)  

 

b)  

 

c) 

 

a)  

 

b)  

 

c) 



a)  

 

b)  

 

c) 



a)  

 

b)  

 

c) 

a)  

 

b)  

 

c) 



Correct answer:

a)  

 

b)  

 

c) 



Explanation:

`

 

If we were to find the roots of this equation we would first set  and attempt to factor:  

 

The trivial solution is , the other possible non-zero solutions will depend on the quadratic factor. A quadratic equation may have either 2 complex roots, 2 real roots, or 1 real root. The conditions determined by a the radical term in the quadratic equation...the discriminate: 

For a quadratic equation, or quadratic factor, of the form  the discriminate is defined as: 

 

For the quadratic factor in this problem we have, 

The discriminate in this case is then,  

 

Now let's recall how the discriminate determines the number and types of roots 

 

 

a) Two Real Roots 

When the discriminate is greater than zero the function has 2 real-valued roots. 

 

b) One Real Root 

When the discriminate is zero, the function has 1 real valued root: 

Apply to the discriminate and solve for 

 

c) Two Complex Roots 

Similarly, for two complex roots solve  , we have: 

 

 

 Summary 

a)  

 

b)  

 

c) 

 

Example Question #1441 : Algebra Ii

 

Suppose the equation above has ,  and  such that . Select which statement must be true about the solutions. 

 

 

 

 

Possible Answers:

There is one real solution and one complex solution.

There are two complex solutions. 

There are two real solutions. 

There are no solutions.

There is one real solution.  

Correct answer:

There is one real solution.  

Explanation:

 

                                         (1)

 

                                             (2)

 

The first step is conceptualize. What kind of equation is equation (1)? If we rearrange it's clearly a quadratic equation. 

 

                             (3)

 

To determine the number of solutions use the discriminate: 

______________________________________________________________

Reminder

Recall that for a quadratic equation   the general formula for the solution in terms of the constant coefficients is given by:  

 

The quantity under the radical is known as the discriminate. If the discriminate is less than zero, there are two complex valued solution. In cases where the discriminate is zero, there is just one real solution . If the discriminate is positive, then there are two real solutions. 

                                                         

 _____________________________________________________________

 

 The discriminate for equation (3) is written: 

 

We are given that  .  Therefore we have: 

 

 

 

The discriminate is therefore zero, meaning there is only one real solution. 

Example Question #141 : Understanding Quadratic Equations

Given , what is the value of the discriminant?

Possible Answers:

Correct answer:

Explanation:

The correct answer is . The discriminant is equal to  portion of the quadratic formula. In this case, "" corresponds to the coefficient of , "" corresponds to the coefficient of , and "" corresponds to . So, the answer is , which is equal to

Example Question #141 : Understanding Quadratic Equations

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

Possible Answers:

-7, 2 real solutions

57, 2 real solutions

-7, 2 imaginary solutions

57, 1 real solution

57, 2 imaginary 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 discriminant 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 : Quadratic Roots

Give the solution set of the equation .

Possible Answers:

Correct answer:

Explanation:

Using the quadratic formula, with :

Example Question #1 : Quadratic Roots

Give the solution set of the equation  .

Possible Answers:

Correct answer:

Explanation:

Using the quadratic formula, with :

Example Question #2 : Quadratic Roots

Write a quadratic equation in the form  with 2 and -10 as its roots.

Possible Answers:

Correct answer:

Explanation:

Write in the form  where p and q are the roots.

Substitute in the roots:

Simplify:

Use FOIL and simplify to get

.

Example Question #4 : Quadratic Roots

Find the roots of the following quadratic polynomial:

Possible Answers:

This quadratic has no real roots.

Correct answer:

Explanation:

To find the roots of this equation, we need to find which values of  make the polynomial equal zero; we do this by factoring. Factoring is a lot of "guess and check" work, but we can figure some things out. If our binomials are in the form , we know  times  will be  and  times  will be . With that in mind, we can factor our polynomial to 

To find the roots, we need to find the -values that make each of our binomials equal zero. For the first one it is , and for the second it is , so our roots are .

Example Question #143 : Understanding Quadratic Equations

Write a quadratic equation in the form  that has  and  as its roots.

Possible Answers:

Correct answer:

Explanation:

1. Write the equation in the form  where  and  are the given roots.

 

2. Simplify using FOIL method.

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