# GMAT Math : DSQ: Solving quadratic equations

## Example Questions

### Example Question #1 : 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.

EITHER statement ALONE is 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.

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

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.

Explanation:

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 #1 : 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:

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.

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

BOTH statements TOGETHER are insufficient to answer the question.

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

Explanation:

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 #151 : Algebra

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

Statement 1:

Statement 2:

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

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.

Explanation:

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 : Solving Quadratic Equations

is a positive integer.

True or false:

Statement 1:  is an even integer

Statement 2:

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 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.

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

Explanation:

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 : Solving Quadratic Equations

is a positive integer.

True or false?

Statement 1:  is an even integer

Statement 2:

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

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.

Explanation:

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 #6 : Solving Quadratic Equations

What are the solutions of  in the most simplified form?