GMAT Math : DSQ: Solving quadratic equations

Study concepts, example questions & explanations for GMAT Math

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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.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question. 

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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

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: 

 

Possible Answers:

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.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient 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.

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 #3 : 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.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 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.

Correct answer:

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 #1 : Solving Quadratic Equations

 is a positive integer.

True or false:

Statement 1:  is an even integer

Statement 2: 

Possible Answers:

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.

Correct answer:

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 #1 : Dsq: Solving Quadratic Equations

 is a positive integer.

True or false?

Statement 1:  is an even integer

Statement 2: 

Possible Answers:

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 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.

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

Correct answer:

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 #1 : Dsq: Solving Quadratic Equations

What are the solutions of  in the most simplified form?

Possible Answers:

Correct answer:

Explanation:

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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