Core Algebra Concepts: What to Know and How They Show Up in Online Homework

Algebra is the part of math where letters stand in for numbers, so you can describe a rule once and use it for every case. Almost every college math course after it, from statistics to calculus, assumes you can move around inside it comfortably. This guide walks through the core concepts in the order they build on each other, with one worked example for each, the mistake that costs the most points, and a note on how online homework systems tend to grade that topic.

Quick answer: Core algebra is a chain of topics that each build on the one before: expressions and order of operations, equations and inequalities, systems, functions and graphs, exponents, polynomials and factoring, quadratics, radicals and rational expressions, and exponential and logarithmic functions, with complex numbers appearing wherever a quadratic has no real solution. A gap early in the chain shows up as repeated mistakes later. Online homework systems also grade the form of an answer, not only its value, so check whether a problem wants a simplified expression, a fraction, or interval notation.

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The Order Algebra Builds In

An arrow of dependency runs down this table: each topic uses the ones listed beside it, so a weak spot early on tends to show up later. If a unit feels impossible, the cause is often a row or two above it.

Topic Builds on Needed for
Expressions and order of operations Arithmetic Every later topic
Linear equations and inequalities Expressions Systems, functions, graphing
Exponent rules Expressions Polynomials, exponential functions
Systems of linear equations Linear equations Modeling, word problems
Functions, domain, and graphs Linear equations All later function work
Polynomials and factoring Exponent rules Quadratics, rational expressions
Quadratic equations Factoring, exponents Parabolas, complex numbers
Radicals and rational expressions Factoring, exponents Calculus-style algebra
Exponential and logarithmic functions Functions, exponents Growth, decay, finance
Complex numbers Quadratics, radicals Algebra 2 and precalculus

Which Course Covers What

Course names and sequences vary by school, so check your syllabus. A common pattern in the United States:

  • Algebra 1: expressions, linear equations and inequalities, graphing lines, systems, exponent rules, and an introduction to polynomials, factoring, and quadratics.
  • Algebra 2: quadratics in depth, polynomial, rational, and radical functions, complex numbers, and exponential and logarithmic functions.
  • Intermediate Algebra: a college-level bridge course that usually covers material similar to Algebra 2.
  • College Algebra: functions in depth, including polynomial, rational, exponential, and logarithmic functions, plus systems and modeling.

For a closer comparison, see Algebra 1 vs Algebra 2.

The Core Concepts, One at a Time

Expressions and Order of Operations

An expression is a phrase with numbers, variables, and operations but no equals sign, such as 3x + 5x – 2. An equation sets two expressions equal. You simplify expressions and solve equations. Simplifying means combining like terms and using the distributive property: 3x + 5x – 2 becomes 8x – 2, and 2(x + 4) becomes 2x + 8. Order of operations says to work through parentheses first, then exponents, then multiplication and division left to right, then addition and subtraction left to right. For example, 6 + 2(5 – 3)2 = 6 + 2(4) = 14.

Costly mistake: sign errors when distributing a negative. -(x – 3) equals -x + 3, not -x – 3.

In online homework: many systems want the fully simplified form, so submitting 3x + 5x – 2 can be marked incorrect even though it equals 8x – 2.

Linear Equations and Inequalities

To solve an equation, do the same thing to both sides until the variable is alone. For 3x + 7 = 22, subtract 7 to get 3x = 15, then divide by 3 to get x = 5. Inequalities work the same way with one extra rule: multiplying or dividing both sides by a negative number flips the inequality sign. Solving -2x > 6 gives x < -3.

Costly mistake: forgetting to flip the sign when dividing by a negative.

In online homework: inequality answers are often entered in interval notation, where x < -3 is written (-∞, -3).

Systems of Linear Equations

A system is two or more equations that share variables, and its solution is the pair of values that satisfies all of them. For 2x + y = 7 and x – y = 2, add the equations to eliminate y: 3x = 9, so x = 3. Substitute back to get y = 1, so the solution is (3, 1). Substitution and graphing reach the same answer, and elimination is usually fastest when the coefficients line up.

Costly mistake: finding one variable and stopping. The solution is an ordered pair.

In online homework: check whether the system wants an ordered pair like (3, 1) or separate boxes for x and y.

Functions, Domain, and Graphs

A function assigns exactly one output to each input, so a set of ordered pairs is a function when no input repeats. For f(x) = x2 – 4x + 3, f(2) = 4 – 8 + 3 = -1. The domain is the set of allowed inputs, and two restrictions cover most problems: you cannot divide by zero, and you cannot take an even root of a negative number in the real numbers. So 1/(x – 3) has a domain of all real numbers except 3, and the square root of (x – 5) has a domain of x ≥ 5.

On a graph, the line y = mx + b has slope m and y-intercept b. The slope between (1, 3) and (4, 9) is (9 – 3)/(4 – 1) = 2, and the line through them is y = 2x + 1.

Costly mistake: giving the domain of a fraction without excluding the value that makes the denominator zero.

In online homework: domain answers usually need interval or set notation, and the difference between a bracket and a parenthesis matters: [5, ∞) includes 5, while (5, ∞) does not.

Exponent Rules

An exponent counts repeated multiplication, so 34 = 3 × 3 × 3 × 3 = 81. The rules are:

Rule Pattern Example
Product am × an = am+n x3 × x4 = x7
Quotient am / an = am-n x7 / x4 = x3
Power of a power (am)n = amn (x2)5 = x10
Negative exponent a-n = 1 / an 2-3 = 1/8
Zero exponent a0 = 1 (a not 0) 50 = 1

Coefficients multiply separately, so (2x3)(3x4) = 6x7.

Costly mistake: -32 equals -9, because the exponent applies before the negative sign, while (-3)2 equals 9.

In online homework: exponents are typically typed with a caret or an on-screen exponent key, and parentheses matter: (x + 3)^2 is not x + 3^2. Our guide on how to type exponents in DeltaMath shows the input side in detail.

Polynomials and Factoring

A polynomial is a sum of terms with whole-number exponents, such as 3x2 + 2x – 5, and its degree is the highest exponent. Factoring rewrites a polynomial as a product. Look for a greatest common factor first, then for patterns. A difference of squares factors as x2 – 9 = (x – 3)(x + 3). A trinomial like x2 – 5x + 6 factors by finding two numbers that multiply to 6 and add to -5, which gives (x – 2)(x – 3).

Costly mistake: stopping early. Check that no factor can be factored further.

In online homework: many systems accept the factors in any order but expect the factoring to be complete.

Quadratic Equations

A quadratic equation has the form ax2 + bx + c = 0 and graphs as a parabola. There are three standard methods, and the quadratic formula and completing the square work on every quadratic:

  • Factoring works when the quadratic factors with whole numbers. x2 – 5x + 6 = 0 becomes (x – 2)(x – 3) = 0, so x = 2 or x = 3.
  • Completing the square rewrites the equation as a perfect square. x2 + 6x + 5 = 0 becomes (x + 3)2 = 4, so x = -1 or x = -5.
  • The quadratic formula is x = (-b ± √(b2 – 4ac)) / (2a).

The discriminant, b2 – 4ac, tells you what to expect before you finish. A positive value gives two real solutions: x2 – 4x – 5 = 0 has a discriminant of 36, so x = 5 or x = -1. Zero gives one repeated real solution. A negative value gives two non-real complex solutions: x2 + 2x + 5 = 0 has a discriminant of -16, so x = -1 ± 2i. On the graph, real solutions are the x-intercepts, and the vertex sits at x = -b/(2a).

Costly mistake: assuming factoring is the only method. If a quadratic does not factor cleanly, use the formula.

In online homework: exact answers are often required, so leave a solution as -1 ± 2i or in radical form unless the problem asks you to round.

Complex Numbers

The imaginary unit i is defined by i2 = -1. Its powers repeat every four: i1 = i, i2 = -1, i3 = -i, i4 = 1. To simplify a large power, divide the exponent by 4 and use the remainder. Since 27 = 4(6) + 3, i27 = i3 = -i. A complex number has the form a + bi. Multiply by treating i like a variable and replacing i2 with -1: (3 + 2i)(1 – i) = 3 – 3i + 2i – 2i2 = 5 – i.

Costly mistake: treating i2 as 1 instead of -1.

In online homework: complex answers are normally expected in a + bi form, with the real part written first.

Radicals and Rational Expressions

To simplify a radical, pull out perfect squares: √50 = √(25 × 2) = 5√2. A rational expression is a fraction containing variables. Simplify it by factoring and canceling common factors: (x2 – 4)/(x – 2) = (x – 2)(x + 2)/(x – 2) = x + 2, with x ≠ 2 because the original expression is undefined there.

Costly mistake: canceling terms instead of factors. You can cancel (x – 2) from a product, but you cannot cancel the x in (x + 3)/x.

In online homework: when a problem asks for simplest form, expect to rationalize denominators and state any excluded values.

Exponential and Logarithmic Functions

In an exponential function such as f(x) = a · bx, the variable is in the exponent. When b > 1 the function grows, and when 0 < b < 1 it decays. Compound interest follows A = P(1 + r)t. A logarithm undoes an exponential: logb(y) = x means bx = y. Since 25 = 32, log2(32) = 5. Logarithms let you solve for an unknown exponent, as in 2x = 32.

Costly mistake: the product rule is log(ab) = log a + log b, but there is no rule that splits log(a + b).

In online homework: ln means the natural log with base e, and log with no base usually means base 10. Check which one the problem uses.

Check Yourself: Seven Practice Problems

  1. Simplify 4(x – 2) + 3x.
  2. Solve 5x – 3 = 2x + 12.
  3. Solve the system x + y = 9 and x – y = 1.
  4. Factor x2 – x – 12.
  5. Solve x2 + 4x + 5 = 0.
  6. Find the domain of √(2x – 6).
  7. Evaluate log3(81).

Answers: (1) 7x – 8. (2) x = 5. (3) x = 5 and y = 4, or (5, 4). (4) (x – 4)(x + 3). (5) The discriminant is -4, so x = -2 ± i. (6) x ≥ 3, or [3, ∞). (7) 4, because 34 = 81.

How Online Homework Systems Grade Algebra

A correct value in the wrong form is one of the most common reasons an algebra answer is marked wrong online. Before you submit, check four things: whether the problem asks for simplified form, whether it wants a fraction or a decimal, whether it expects interval notation or an inequality, and whether parentheses and exponents were typed the way the system reads them. If you are working in one of these systems, we have platform-specific guides:

For free practice, Khan Academy’s algebra course covers most of the topics above, and the Desmos graphing calculator is a quick way to check a graph, an intercept, or a solution.

Falling Behind in an Algebra Course?

Algebra builds on itself, so one missed unit can slow down everything after it, especially in a self-paced online course with a deadline. Finish My Math Class has been helping students through Algebra 1, Algebra 2, and College Algebra since 2016, and our work is backed by an A/B Grade Guarantee. You can read client testimonials or get a free quote.

Frequently Asked Questions

What is the difference between an expression and an equation?

An expression is a phrase without an equals sign, such as 2x + 3. An equation sets two expressions equal, such as 2x + 3 = 7. You simplify expressions and solve equations.

How do I know if an equation is linear?

An equation is linear when each variable appears only to the first power, is not multiplied by another variable, and is not under a root or in a denominator. 2x + 3 = 7 and y = -5x + 1 are linear. Anything with x^2, the square root of x, or xy is not, and its graph is not a straight line.

What is the domain of a function?

The domain is the set of inputs for which the function is defined. For real-valued functions, exclude any value that makes a denominator zero or puts a negative number under an even root. The domain of 1/(x – 3) is all real numbers except 3, and the domain of the square root of (x – 5) is x >= 5.

What is a polynomial function?

A polynomial function is a sum of terms, each a constant times a variable raised to a whole-number power, such as f(x) = 4x^3 – 2x^2 + x – 7. The highest exponent is the degree, so this example has degree 3.

What is i in algebra?

The letter i is the imaginary unit, defined by i^2 = -1. It lets you solve equations that have no real solutions. For example, x^2 + 1 = 0 has the solutions i and -i.

How do I simplify powers of i?

Powers of i repeat every four: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. Divide the exponent by 4 and use the remainder. For i^27, 27 divided by 4 leaves a remainder of 3, so i^27 = i^3 = -i.

Do I need algebra before calculus?

Yes. Calculus assumes fluency with functions, exponents, factoring, and solving equations, and usually with precalculus topics such as logarithms and trigonometry. Many schools require College Algebra, Precalculus, or a placement score first, so check your catalog.

Why do colleges require algebra for non-STEM majors?

Algebra is a general education requirement at many colleges because it builds quantitative reasoning: working with unknowns, reading graphs, and checking whether a result makes sense. Requirements vary by school and program, so confirm yours in the catalog.