Algebra Equations | Formula, Types & Examples

Here’s a complete guide to algebraic equations: definitions, common types, key formulas, and worked examples.

Definition

An algebraic equation is a statement that two algebraic expressions are equal, typically containing one or more variables (unknowns). Solving an equation means finding all values of the variable(s) that make the equality true.

Key formulas and facts

• Linear equation (one variable): $ax+b=0 \Rightarrow x=-\\frac{b}{a}$
• Quadratic standard form: $ax^2+bx+c=0$
• Quadratic formula: $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
• Discriminant: $\Delta=b^2-4ac$ (if $\Delta>0$ two real roots; $\Delta=0$ one real root; $\Delta<0$ two complex roots)
• Sum/difference of squares: $a^2-b^2=(a-b)(a+b)$
• Sum/difference of cubes: $a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)$
• Completing the square: $ax^2+bx+c=a\Big(x+\frac{b}{2a}\Big)^2-\frac{b^2-4ac}{4a}$
• For exponential $a^{x}=b \Rightarrow x=\log_a b = \frac{\ln b}{\ln a}$
• Log rules: $\log(AB)=\log A+\log B$, $\log\left(\frac{A}{B}\right)=\log A-\log B$, $\log(A^k)=k\log A$

Common types & how to solve them (with examples)

1) Linear equations

Example: Solve $3x-7=8$
Steps:

1. Add 7: $3x=15$
2. Divide by 3: $x=5$

2) Quadratic equations

Example A (factorable): Solve $x^2-3x-4=0$
Steps:

1. Factor: $(x-4)(x+1)=0$
2. Solutions: $x=4,\ -1$

Example B (use quadratic formula): Solve $2x^2-4x-3=0$
Steps:

1. Apply formula: $$x=\frac{-(-4)\pm\sqrt{(-4)^2-4\cdot2\cdot(-3)}}{2\cdot2}=\frac{4\pm\sqrt{16+24}}{4}=\frac{4\pm\sqrt{40}}{4}$$
2. Simplify: $x=\frac{4\pm2\sqrt{10}}{4}=\frac{2\pm\sqrt{10}}{2}$

3) Systems of linear equations (two variables)

Example: Solve
\begin{cases}
2x+3y=6\\
x-y=1
\end{cases}
Steps:

1. From second: $x=y+1$. Substitute into first: $2(y+1)+3y=6\Rightarrow5y+2=6\Rightarrow y=\frac{4}{5}$
2. Then $x=y+1=\frac{4}{5}+1=\frac{9}{5}$

4) Rational equations

Example: Solve $\frac{1}{x}+\frac{2}{x+1}=3$ (note: $x\neq0,-1$)
Steps:

1. Multiply both sides by $x(x+1)$: $(x+1)+2x=3x(x+1)$
2. Simplify: $3x+1=3x^2+3x\Rightarrow3x^2-1=0$
3. Solve: $x^2=\frac{1}{3}\Rightarrow x=\pm\frac{1}{\sqrt{3}}$ (both valid if not excluded by domain)

5) Radical equations

Example: Solve $\sqrt{2x+1}=x-1$ (domain: $x-1\ge0\Rightarrow x\ge1$)
Steps:

1. Square both sides: $2x+1=(x-1)^2=x^2-2x+1$
2. Rearrange: $0=x^2-4x\Rightarrow x(x-4)=0$
3. Candidates: $x=0$ or $x=4$. Check domain and original: $x=0$ invalid (domain), $x=4$ valid.

Always check for extraneous roots after squaring.

6) Exponential and logarithmic equations

Example: Solve $2^{x+1}=8$
Steps:

1. $8=2^3$, so $2^{x+1}=2^3\Rightarrow x+1=3\Rightarrow x=2$

Example: Solve $5^x=20$
Steps:

1. Take logs: $x=\frac{\ln 20}{\ln 5}$

7) Absolute value equations

Example: $|2x-3|=5$
Steps:

1. Solve $2x-3=5 \Rightarrow x=4$ and $2x-3=-5\Rightarrow x=-1$

Tips and best practices

• Always state domain restrictions (no division by zero, radicand nonnegative for real square roots).
• Check solutions in the original equation especially for rational and radical cases.
• Use factoring, completing the square, or the quadratic formula depending on the form.
• For systems, use substitution, elimination, or matrices depending on size/complexity.

If you want, I can provide more worked examples for any specific type (e.g., cubic, inequalities, systems with 3 variables, or word-problem modeling). Which type would you like next?