Solving Linear and Quadratic Equations

1.1 Linear Equations

Definition and Overview

Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. The general form of a linear equation in one variable is:

• Standard Form: $ax + b = 0$, where a and b are constants and a \neq 0.

Solving a linear equation involves isolating the variable on one side of the equation.

Solving Simple Linear Equations

• Step 1: Simplify both sides of the equation if necessary (combine like terms, remove parentheses).

• Step 2: Use addition or subtraction to isolate terms containing the variable on one side.

• Step 3: Use multiplication or division to solve for the variable.

Example 1: $2x + 3 = 9$

• Subtract 3 from both sides: $2x = 6$

• Divide both sides by 2: $x = 3$

Example 2: $x - 5 = 3$

• Add 5 to both sides: $x = 8$

Solving Linear Equations with Fractions

• Clear denominators by multiplying both sides by the least common denominator (LCD).

• Proceed as with standard linear equations.

Example: $\frac{x}{4} + 3 = \frac{2x - x}{4}$

• Combine like terms: $\frac{x}{4} + 3 = \frac{x}{4}$

• Subtract $\frac{x}{4}$ from both sides: $3 = 0$ (No solution in this case)

Solving Linear Equations with Variables on Both Sides

• Move all variable terms to one side and constants to the other.

• Simplify and solve for the variable.

Example: $8x + 10x = 22 + 4x$

• Combine like terms: $18x = 22 + 4x$

• Subtract $4x$ from both sides: $14x = 22$

• Divide by 14: $x = \frac{22}{14} = \frac{11}{7}$

1.6 Quadratic Equations

Definition and Overview

A quadratic equation is a second-degree polynomial equation in one variable, generally written as:

• Standard Form: $ax^2 + bx + c = 0$, where a \neq 0.

Solving Quadratic Equations by Factoring

• Rewrite the equation in standard form.

• Factor the quadratic expression if possible.

• Set each factor equal to zero and solve for the variable.

Example: $x^2 - 6x + 8 = 0$

• Factor: $(x - 2)(x - 4) = 0$

• Set each factor to zero: $x - 2 = 0$ or $x - 4 = 0$

• Solutions: $x = 2$ or $x = 4$

Solving Quadratic Equations by the Quadratic Formula

• If factoring is not possible, use the quadratic formula:

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

• Where $a$, $b$, and $c$ are coefficients from $ax^2 + bx + c = 0$.

Example: $x^2 + 3x - 18 = 0$

• Here, $a = 1$, $b = 3$, $c = -18$

• Plug into the formula:

$x = \frac{ -3 \pm \sqrt{3^2 - 4(1)(-18)} }{2(1)} = \frac{ -3 \pm \sqrt{9 + 72} }{2} = \frac{ -3 \pm \sqrt{81} }{2} = \frac{ -3 \pm 9 }{2}$

• So $x = 3$ or $x = -6$

Solving Quadratic Equations by Completing the Square

• Move constant term to the other side.

• Add the square of half the coefficient of $x$ to both sides.

• Write the left side as a squared binomial and solve for $x$.

Example: $x^2 + 4x + 1 = 0$

• Move 1: $x^2 + 4x = -1$

• Add $(\frac{4}{2})^2 = 4$ to both sides: $x^2 + 4x + 4 = 3$

• Write as $(x + 2)^2 = 3$

• Take square root: $x + 2 = \pm \sqrt{3}$

• So $x = -2 \pm \sqrt{3}$

Key Properties and Tips

• Always check for extraneous solutions, especially when dealing with rational equations.

• Quadratic equations can have two real solutions, one real solution, or two complex solutions depending on the discriminant $b^2 - 4ac$.

• Linear equations always have one solution unless the equation is inconsistent (no solution) or dependent (infinitely many solutions).

Summary Table: Methods for Solving Equations

Equation Type	Standard Form	Solution Methods
Linear	$ax + b = 0$	Isolate variable, clear fractions, combine like terms
Quadratic	$ax^2 + bx + c = 0$	Factoring, Quadratic Formula, Completing the Square