### All GMAT Math Resources

## Example Questions

### Example Question #11 : Solving Equations

Solve for :

**Possible Answers:**

The equation has no solution.

**Correct answer:**

### Example Question #12 : Equations

Give all real solutions of the following equation:

**Possible Answers:**

The equation has no solution.

**Correct answer:**

By substituting and, subsequently, , this equation be rewritten as a quadratic equation, and solved as such:

We are looking to factor the quadratic expression as , replacing the two question marks with integers with a product of 9 and a sum of ; these integers are .

Substitute back:

These factors can themselves be factored as the difference of squares:

Set each factor to zero and solve:

The solution set is .

### Example Question #13 : Equations

Solve for :

**Possible Answers:**

The equation has no solution.

**Correct answer:**

Substitute , and, subsequently, , to rewrite this equation as quadratic, then solve by factoring.

We can rewrite the quadratic expression as , where the question marks are replaced with integers whose product is 12 and whose sum is ; these integers are .

Set each factor to zero and solve for ; then substitute back and solve for :

The solution set, which can be confirmed by substitution, is .

### Example Question #11 : Equations

Find all real solutions to the following equation:

**Possible Answers:**

The equation has no solution.

**Correct answer:**

This can be best solved by substituting , and, subsequently, , then solving the resulting quadratic equation.

Factor the expression on the left by finding two integers whose product is 12 and whose sum is :

Set each linear binomial factor to 0, solve separately for , and substitute back:

or

### Example Question #1411 : Problem Solving Questions

The period of a pendulum - that is, the time it takes for the pendulum to swing once and back - varies directly as the square root of its length.

The pendulum of a giant clock is 18 meters long and has period 8.5 seconds. If the pendulum is lengthened to 21 meters, what will its period be, to the nearest tenth of a second?

**Possible Answers:**

**Correct answer:**

The variation equation for this situation is

Set , and solve for ;

### Example Question #11 : Solving Equations

Which of these expressions is equal to ?

**Possible Answers:**

**Correct answer:**

### Example Question #17 : Equations

Solve for :

**Possible Answers:**

**Correct answer:**

First, isolate the absolute value expression on one side:

Rewrite as a compound sentence:

Solve each separately:

or

The solution set is

### Example Question #12 : Solving Equations

Solve for :

**Possible Answers:**

**Correct answer:**

First, isolate the absolute value expression on one side:

Rewrite as a compound sentence:

Solve each separately:

or

The solution set is .

### Example Question #1413 : Problem Solving Questions

Give the solution set of the equation:

**Possible Answers:**

The equation has no solution.

**Correct answer:**

Write this as a compound equation and solve each separately.

This gives us three possible solutions - . We check all three.

This is a false statement so we can eliminate as a false "solution".

2 is a solution.

3 is a solution.

The solution set is .

### Example Question #11 : Solving Equations

Solve for :

**Possible Answers:**

The equation has no solution

**Correct answer:**

Since and , replace, and use the exponent rules:

Set the exponents equal to each other and solve for :

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