Algebra Fundamentals

Cheatsheet Content

### Basic Properties - **Commutative Property:** - Addition: $a + b = b + a$ - Multiplication: $a \cdot b = b \cdot a$ - **Associative Property:** - Addition: $(a + b) + c = a + (b + c)$ - Multiplication: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ - **Distributive Property:** $a(b + c) = ab + ac$ - **Identity Property:** - Addition: $a + 0 = a$ - Multiplication: $a \cdot 1 = a$ - **Inverse Property:** - Addition: $a + (-a) = 0$ - Multiplication: $a \cdot (1/a) = 1$ (for $a \neq 0$) ### Exponents - **Product Rule:** $x^m \cdot x^n = x^{m+n}$ - **Quotient Rule:** $x^m / x^n = x^{m-n}$ - **Power Rule:** $(x^m)^n = x^{mn}$ - **Power of a Product:** $(xy)^n = x^n y^n$ - **Power of a Quotient:** $(x/y)^n = x^n / y^n$ - **Zero Exponent:** $x^0 = 1$ (for $x \neq 0$) - **Negative Exponent:** $x^{-n} = 1/x^n$ - **Fractional Exponent:** $x^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m$ ### Radicals - **Definition:** If $x^n = a$, then $x = \sqrt[n]{a}$ - **Product Rule:** $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ - **Quotient Rule:** $\sqrt[n]{a/b} = \sqrt[n]{a} / \sqrt[n]{b}$ - **Simplifying:** $\sqrt{x^2} = |x|$ - **Converting to Exponents:** $\sqrt[n]{x^m} = x^{m/n}$ ### Factoring Formulas - **Difference of Squares:** $a^2 - b^2 = (a-b)(a+b)$ - **Sum of Cubes:** $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$ - **Difference of Cubes:** $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$ - **Perfect Square Trinomials:** - $a^2 + 2ab + b^2 = (a+b)^2$ - $a^2 - 2ab + b^2 = (a-b)^2$ ### Quadratic Equations - **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 distinct real solutions - If $\Delta = 0$, one real solution (repeated) - If $\Delta ### Linear Equations - **Slope-Intercept Form:** $y = mx + b$ - $m$: slope, $b$: y-intercept - **Point-Slope Form:** $y - y_1 = m(x - x_1)$ - **Standard Form:** $Ax + By = C$ - **Slope ($m$):** - Given two points $(x_1, y_1)$ and $(x_2, y_2)$: $m = \frac{y_2 - y_1}{x_2 - x_1}$ - Parallel lines: $m_1 = m_2$ - Perpendicular lines: $m_1 \cdot m_2 = -1$ (or $m_1 = -1/m_2$) ### Inequalities - **Properties:** - Adding/Subtracting a number: $a 0$) - Multiplying/Dividing by a negative number: $a bc$ (if $c 0$) - $|x| > k \iff x k$ (for $k > 0$) ### Logarithms - **Definition:** $y = \log_b x \iff b^y = x$ - **Product Rule:** $\log_b(xy) = \log_b x + \log_b y$ - **Quotient Rule:** $\log_b(x/y) = \log_b x - \log_b y$ - **Power Rule:** $\log_b(x^p) = p \cdot \log_b x$ - **Change of Base Formula:** $\log_b x = \frac{\log_c x}{\log_c b}$ - **Special Logarithms:** - **Common Log:** $\log x = \log_{10} x$ - **Natural Log:** $\ln x = \log_e x$ - **Inverse Properties:** - $b^{\log_b x} = x$ - $\log_b (b^x) = x$