# GMAT Math : Solving equations

## Example Questions

### Example Question #31 : Solving Equations

Solve for . Give all real solutions:

The equation has no real solution.

Explanation:

One way is to substitute , and, subsequently,

Set each binomial to 0 and solve separately:

or

Since no real number has  as its principal square root, this yields no solution.

The only solution is .

### Example Question #31 : Solving Equations

Solve for , giving all solutions, real and imaginary:

Explanation:

Factor the expression:

Rewrite:

or

### Example Question #32 : Solving Equations

Solve for :

The equation has no solution.

Explanation:

, so the equation can be rewritten as follows:

Set the exponents equal to each other:

### Example Question #31 : Solving Equations

Daniel has  candy bars. Andy has three more candy bars than the double of Daniel's candy bars.

How many candy bars does Andy have?

Explanation:

Let A be the number of Andy's candy bars and D be the number of Daniel's candy bars.

We start by setting up the equation:

and

So

### Example Question #33 : Solving Equations

What is the value of ?

Explanation:

To find the value of x we need to isolate x on one side of the equation and the rest of the numbers on the other side.

First, we multiply what is in the denominator on the left had side by the numerators on both sides.

Then we distribute the 5 to both terms in the binomial. Doing this we get a zero in the exponent.

Anything raised to the zero just becomes one.

From here we subtract 0.5 from each side to solve for x.

### Example Question #34 : Solving Equations

Define  . Which of the following would be a valid alternative way of expressing the definition of ?

Explanation:

By definition:

If , then  ,and subsequently,

If , then  ,and subsequently,

### Example Question #341 : Algebra

Which of the following expressions is equal to  ?

Explanation:

, so , and .

### Example Question #36 : Solving Equations

A factory makes barrels of the same shape but different sizes; the amount of water they hold varies directly as the cube of their height. The four-foot-high barrel holds 20 gallons of water; how much water would the six-foot-high barrel hold?

Explanation:

Let  be the height of a barrel and  be its volume. Since  varies directly as the cube of , the variation equation is

for some constant of variation .

We find  by substituting  from the smaller barrels:

Then the variation equation is:

Now we can substitute  to find the volume of the larger barrel:

The larger barrel holds  gallons.

Solve for .

Explanation:

### Example Question #31 : Equations

To convert Celsius temperature  to the equivalent in Fahrenheit temperature , use the formula

To the nearest tenth of a degree, convert  to degrees Fahrenheit.