TABLE OF CONTENTS

## Baruch MTH 1030 Final Overview

MTH 1030 is an algebra course in Baruch College. This course covers most basic quantitative courses at the college, including linear equations, rates of change, rational expressions, circles, functions and their graphs, inverse functions, exponential and logarithmic functions, the geometric series, an introduction to annuities, non-linear systems of equations and related applications. Therefore, the students are required to master TI 89 or TI 92 graphical calculator for this course.

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## Baruch MTH 1030 Final Topic 1. Exponents and Polynomials.

### Rule 1.

Turn both parts of the equation to the same base.

### Rule 2.

Set exponents on both sides equal to each other and solve for .

## MTH 1030, Sample Final FA16, Problem 10

We need to solve for x from the following equation

### Step 1.

Get the same base on both sides:

### Step 2.

Set exponents on both sides equal to each other and solve for :

The answer is .

## Baruch MTH 1030 Final Topic 2. Circle.

Equation of a circle with radius and origin is . Complete the square for and to turn the equation into the form .

## MTH 1030, Sample Final FA18, Problem 2

In Problem 2 of the Final Test FA18 we need to find the center and the radius of the circle

### Step 1.

Complete the squares for and :

The center is and the radius is

The answer is .

## Baruch MTH 1030 Final Topic 3. Rationalizing.

Rationalizing is often connected to the concept of conjugate.

The conjugate of is .

The conjugate of is .

## MTH 1030, Sample Final FA16, Problem 3

In Problem 3 of the Final Test FA16 we need to rationalize the denominator of the following fraction

### Step 1.

It is clear that in order to rationalize the denominator, we need to identify the conjugate of the denominator and multiply both top and bottom by that very conjugate.

Denominator is .

The conjugate to the denominator is . Letâ€™s multiply both top and bottom by :

### Step 2.

Multiply out both top and bottom and simplify:

The answer is .

## Baruch MTH 1030 Final Topic 4. Complex Numbers.

Problems containing fractions with complex numbers are often solved in similar fashion described in the previous topic of rationalizing.

The conjugate of is .

The conjugate of is .

Also, letâ€™s not forget that

## MTH 1030, Sample Final FA16, Problem 4

In Problem 4 of the Final Test FA16 we need to rationalize the denominator of the following fraction

### Step 1.

It is clear that in order to rationalize the denominator, we need to identify the conjugate of the denominator and multiply both top and bottom by that very conjugate.

Denominator is .

The conjugate to the denominator is . Letâ€™s multiply both top and bottom by :

### Step 2.

Multiply out both top and bottom and simplify:

The answer is .

## Baruch MTH 1030 Final Topic 5. Domain and Range.

Domain can be found by examining the denominator of a fraction.

### Step 1.

Set the denominator to and solve for

### Step 2.

Exclude all solutions from Step 1. The remainder of values will be included in the domain.

## MTH 1030, Sample Final FA16, Problem 1

In Problem 1 of the Final Test FA16 we need to find the domain of the fraction

### Step 1.

Letâ€™s set the denominator to and solve for :

and

### Step 2.

We need to know exclude x=0 and x=4 from the set of all values.

Thus, the domain is .

The answer is .

## Baruch MTH 1030 Final Topic 6. Completing the Square.

To solve a quadratic equation by completing the square follow these steps:

### Step 1.

Complete the square

### Step 2.

Take square root of both sides and solve for

## MTH 1030, Sample Final FA18, Problem 12

In Problem 12 of the Final Test FA18 we need to solve the quadratic equation by completing the square:

### Step 1.

Complete the square first

### Step 2.

Take square root of both sides and solve for

The answer is .

## Baruch MTH 1030 Final Topic 7. Radical Equations.

To solve a radical equation, follow these steps:

### Step 1.

Move radical part of the equation to one side and everything else to the other side

### Step 2.

Square both sides of the equation, move everything to one side and solve for

### Step 3.

Make sure to double check all answers from Step 2 by plugging into the original equation

## MTH 1030, Sample Final FA18, Problem 5

In Problem 5 of the Final Test FA18 we need to solve the radical equation :

### Step 1.

The equation is already properly prepared. The radical part of the equation is already on the left-hand side and the rest is on the right-hand side

### Step 2.

Letâ€™s now square both sides of the equation and solve for :

and .

### Step 3.

It is time to double check whether and are qualified answers.

When . Thus, works.

When . Thus, does not work.

The answer is .

## Baruch MTH 1030 Final Topic 8. Rational Exponents.

Rational exponential expressions become easy with the following rules:

### Rule 1:

### Rule 2:

### Rule 3:

### Rule 4:

## MTH 1030, Sample Final FA16, Problem 11

In Problem 11 of the Final Test FA16 we need to simplify :

### Step 1.

Letâ€™s simplify the expression within the brackets first:

### Step 2.

Letâ€™s now incorporate the outside exponent:

The answer is .

## Baruch MTH 1030 Final Topic 9. Logarithmic Exponents.

Logarithmic equations are also quite easy with the following rules:

### Rule 1:

### Rule 2:

### Rule 3:

## MTH 1030, Sample Final FA18, Problem 18

In Problem 18 of the Final Test FA18 we need to solve for from the following equation:

### Step 1.

Letâ€™s combine logarithms together:

### Step 2.

We can now set expressions within logarithms equal to one another and solve for :

and .

### Step 3.

Letâ€™s test both answers by plugging them back into the original equation:

:

Thus, works.

:

. There are no solutions, since expression inside the logarithm must be always greater than .

Thus, must be rejected.

The final answer is .

## Baruch MTH 1030 Final Topic 10. Inverse Functions.

The following two steps will help you solve any Inverse Function problem.

### Step 1.

and x and y values need to be inter-changed.

### Step 2.

Solve for

Letâ€™s show how inverse problems can be easily solved with the aforementioned two steps.

## MTH 1030, Sample Final FA16, Problem 23

The inverse of should start with the realization that

Re-writing the equation always helps with the inverse functions.

### Step 1.

Replacing and values results in the following:

### Step 2.

Solve for :

### Step 3.

Letâ€™s now plug in :

The answer is .