Algebra Cheat Sheet: Rules, Formulas, and Problem-Solving Steps

Properties and order of operations

Use grouping symbols first, then exponents, multiplication and division from left to right, and addition and subtraction from left to right. Distribution multiplies every term inside a group.

Like terms have identical variable parts. Their coefficients can be combined; unlike terms cannot.

• Commutative: $a+b=b+a$ and $ab=ba$
• Associative: $(a+b)+c=a+(b+c)$
• Distributive: $a(b+c)=ab+ac$
• Identity: $a+0=a$ and $a\cdot 1=a$
• Inverse: $a+(-a)=0$ and $a\cdot\frac{1}{a}=1$ when $a\ne 0$

Solving equations and inequalities

Simplify each side, move variable terms to one side, isolate the variable using inverse operations, and check in the original equation. Equality is preserved when the same valid operation is applied to both sides.

When multiplying or dividing an inequality by a negative number, reverse the inequality sign. Show this explicitly rather than relying on memory alone.

• Clear fractions by multiplying every term by the least common denominator
• Distribute before combining like terms
• Check for identities with infinitely many solutions
• Check for contradictions with no solution
• Use interval or inequality notation as requested

Exponent and radical rules

Exponent rules apply to common bases in multiplication and division. A negative exponent indicates a reciprocal; it does not make the value negative.

Even roots require attention to domain in real-number problems. Squaring both sides can introduce extraneous solutions, so always check.

• $a^m a^n=a^{m+n}$
• $\dfrac{a^m}{a^n}=a^{m-n}$ for $a\ne 0$
• $(a^m)^n=a^{mn}$
• $(ab)^n=a^n b^n$
• $a^{-n}=\dfrac{1}{a^n}$
• $a^0=1$ for $a\ne 0$
• $a^{1/n}=\sqrt[n]{a}$

Lines and systems

Slope measures vertical change divided by horizontal change. In $y=mx+b$, $m$ is slope and $b$ is the y-intercept. In point-slope form, $y-y_1=m(x-x_1)$.

Solve systems by graphing, substitution, or elimination. A single intersection gives one solution; parallel distinct lines give none; equivalent lines give infinitely many.

• Slope: $m=\dfrac{y_2-y_1}{x_2-x_1}$
• Slope-intercept form: $y=mx+b$
• Point-slope form: $y-y_1=m(x-x_1)$
• Parallel nonvertical lines: $m_1=m_2$
• Perpendicular nonvertical lines: $m_1m_2=-1$

Factoring and quadratics

Before factoring, take out the greatest common factor. Then look for patterns such as a difference of squares or a trinomial. Factoring is useful because a product equals zero only when at least one factor equals zero.

Quadratic solutions can also be found by completing the square or using the quadratic formula. The discriminant $b^2-4ac$ indicates the number and type of real solutions.

• Difference of squares: $a^2-b^2=(a-b)(a+b)$
• Perfect square: $a^2+2ab+b^2=(a+b)^2$
• Zero-product property: if $ab=0$, then $a=0$ or $b=0$
• Quadratic formula: $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
• Axis of symmetry: $x=-\dfrac{b}{2a}$

Functions and rational expressions

Function notation $f(x)$ names an output for input $x$; it does not mean $f$ times $x$. Domain restrictions exclude inputs that make a denominator zero or an even root negative in real-number settings.

Cancel factors, not terms. Factor numerators and denominators completely before simplifying a rational expression, and preserve the original restrictions.

• Evaluate by substituting with parentheses
• For composition, evaluate the inside function first
• For inverses, interchange input and output and solve
• State excluded denominator values
• Check transformed equations for extraneous solutions

A reliable problem-solving checklist

Identify what the problem asks, list known information, choose a representation, and estimate the kind of answer expected. After solving, substitute or interpret the result in context.

The most common preventable errors come from dropped negative signs, incomplete distribution, illegal cancellation, and checking only the final arithmetic.