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## Example Questions

### Example Question #1 : Dsq: Solving Quadratic Equations

Consider the equation

How many real solutions does this equation have?

Statement 1: There exists two different real numbers such that and

Statement 2: is a positive integer.

**Possible Answers:**

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

can be rewritten as

If Statement 1 holds, then the equation can be rewritten as . This equation has solution set , which comprises two real numbers.

If Statement 2 holds, the discriminant is positive, being the sum of a nonnegative number and a positive number; this makes the solution set one with two real numbers.

### Example Question #2 : Dsq: Solving Quadratic Equations

Let be two positive integers. How many real solutions does the equation have?

Statement 1: is a perfect square of an integer.

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

**Correct answer:**

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

The number of real solutions of the equation depends on whether discriminant is positive, zero, or negative; since , this becomes .

If we only know that is a perfect square, then we still need to know to find the number of real solutions. For example, let , a perfect square. Then the discriminant is , which can be positive, zero, or negative depending on .

But if we know , then the discriminant is

Therefore, has one real solution.

### Example Question #3 : Dsq: Solving Quadratic Equations

Does the solution set of the following quadratic equation comprise two real solutions, one real solution, or one imaginary solution?

Statement 1:

Statement 2:

**Possible Answers:**

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

The sign of the discriminant of the quadratic expression answers this question; here, the discriminant is

,

or

If we assume Statement 1 alone, this expression becomes

Since we can assume is nonzero, . This makes the discriminant positive, proving that there are two real solutions.

If we assume Statement 2 alone, this expression becomes

The sign of can vary.

Case 1:

Then

giving the equation two imaginary solutions.

Case 2:

Then

giving the equation two real solutions.

Therefore, Statement 1, but not Statement 2, is enough to answer the question.

### Example Question #4 : Dsq: Solving Quadratic Equations

is a positive integer.

True or false:

Statement 1: is an even integer

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

The quadratic expression can be factored as , replacing the question marks with integers whose product is 8 and whose sum is . These integers are , so the equation becomes:

Set each linear binomial to 0 and solve:

Therefore, for the statement to be true, either or . Each of Statement 1 and Statement 2, taken alone, leaves other possible values of . Taken together, however, they are enough, since the only two positive even integers less than 6 are 2 and 4.

### Example Question #5 : Dsq: Solving Quadratic Equations

is a positive integer.

True or false?

Statement 1: is an even integer

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

The two statements together are insufficient. If both are assumed, then can be 2, 4, or 6.

If the statement is true:

But if the statement is false:

### Example Question #2 : Dsq: Solving Quadratic Equations

What are the solutions of in the most simplified form?

**Possible Answers:**

**Correct answer:**

This is a quadratic formula problem. Use equation . For our problem, Plug these values into the equation, and simplify:

. Here, we simplified the radical by

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