Basic Equations & Algebra

Introduction

Mathematics is the language of the universe. In this lesson, we'll explore fundamental algebraic concepts using properly formatted mathematical notation that makes complex expressions easy to read and understand.

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Linear Equations

A linear equation is the simplest type of equation. It has the general form:

$ax + b = 0$

where $a$ and $b$ are constants, and $x$ is the variable we want to solve for.

Solution:

$x = -\frac{b}{a}$

Example

Solve $3x + 6 = 0$:

$x = -\frac{6}{3} = -2$

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Quadratic Equations

A quadratic equation has the form:

$ax^2 + bx + c = 0$

The solutions are given by the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The expression under the square root, $\Delta = b^2 - 4ac$, is called the discriminant. It tells us:

• If $\Delta > 0$: two real solutions
• If $\Delta = 0$: one repeated solution
• If $\Delta < 0$: no real solutions (complex solutions)

Example

Solve $x^2 - 5x + 6 = 0$:

$a = 1, \quad b = -5, \quad c = 6$

$\Delta = (-5)^2 - 4(1)(6) = 25 - 24 = 1$

$x = \frac{5 \pm \sqrt{1}}{2} = \frac{5 \pm 1}{2}$

$x_1 = 3, \quad x_2 = 2$

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Exponents and Powers

The laws of exponents are fundamental to algebra:

Fractional Exponents

$a^{1/n} = \sqrt[n]{a}$

$a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$

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Logarithms

The logarithm is the inverse of exponentiation. If $b^y = x$, then:

$\log_b(x) = y$

Common Logarithms

Natural logarithm (base $e$):

$\ln(x) = \log_e(x)$

Common logarithm (base 10):

$\log(x) = \log_{10}(x)$

Logarithm Properties

$\log_b(xy) = \log_b(x) + \log_b(y)$

$\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$

$\log_b(x^n) = n \cdot \log_b(x)$

Change of base formula:

$\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$

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Systems of Equations

A system of linear equations can be written in matrix form:

$A\mathbf{x} = \mathbf{b}$

where:

$A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix}$

For a 2×2 system, the solution is:

$x_1 = \frac{\det(A_1)}{\det(A)}, \quad x_2 = \frac{\det(A_2)}{\det(A)}$

where $\det(A) = a_{11}a_{22} - a_{12}a_{21}$.

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Practice Questions

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• Linear equations have the form $ax + b = 0$ with solution $x = -b/a$
• Quadratic equations use the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Exponent laws simplify complex power expressions
• Logarithms are the inverse of exponentiation
• Systems of equations can be solved using matrices and determinants