Exam Prep

Complex Numbers Operations and Simplification

• Definition and Form of Complex Numbers:

  • Standard form is $a + bi$, where $a$ represents the real part and $bi$ represents the imaginary part.

• Addition and Subtraction of Complex Numbers:

  • Problem Example: Subtracting complex expressions: $(5 + 5i) - (-3 - i)$.

  • Step-by-step distribution:

  • Distribute the minus sign across the second complex term: $-(-3) = +3$ and $-(-i) = +i$.

  • Rewrite the full expression: $5 + 5i + 3 + i$.

  • Combine like terms:

  • Combine real parts: $5 + 3 = 8$.

  • Combine imaginary parts ($i$ is treated as $1i$): $5i + 1i = 6i$.

  • Final Result: $8 + 6i$.

  • Test Strategy Tip: When completing assessments online, record the question number on scratch paper and begin working immediately rather than wasting time copying problem statements verbatim.

• Division of Complex Expressions with Radicals:

  • Problem Example: Simplifying and dividing complex radical fractions: \n\frac{-8 + \sqrt{-12}}{40}\n

  • Step 1: Simplify the radical in the numerator:

  • \n\sqrt{-12} = \sqrt{-1 \times 4 \times 3} = 2i\sqrt{3}\n

  • The expression becomes \n\frac{-8 + 2i\sqrt{3}}{40}\n.

  • Step 2: Separate into distinct individual fractions:

  • Based on the fraction addition rule \n\frac{a + b}{c} = \frac{a}{c} + \frac{b}{c}\n, divide each numerator component by the common denominator $40$:

  • \n\frac{-8}{40} + \frac{2i\sqrt{3}}{40}\n

  • Step 3: Reduce each fraction independently:

  • Simplify real fraction: \n\frac{-8}{40} = -\frac{1}{5}\n

  • Simplify imaginary fraction: \n\frac{2}{40} = \frac{1}{20}\n, giving \n\frac{i\sqrt{3}}{20}\n or \n\frac{\sqrt{3}}{20}i\n

  • Final $a + bi$ Form: \n-\frac{1}{5} + \frac{\sqrt{3}}{20}i\n or \n-\frac{1}{5} + \frac{i\sqrt{3}}{20}\n

  • Placement Rules for Imaginary Unit $i$:

  • $i$ can be placed directly adjacent to the fraction or behind the radical (e.g., \n\frac{\sqrt{3}}{20}i\n).

  • $i$ must NEVER be placed under the radical sign.

  • $i$ must NEVER be placed in the denominator of a reduced expression.

• Division of Complex Binomials using Conjugates:

  • Problem Example: Dividing two complex binomials: \n\frac{5 + i}{5 - i}\n

  • Step 1: Identify and multiply by the complex conjugate of the denominator:

  • The complex conjugate of $a - bi$ is $a + bi$. For $5 - i$, the conjugate is $5 + i$.

  • Multiply both numerator and denominator by $5 + i$:

  • \n\frac{(5 + i)(5 + i)}{(5 - i)(5 + i)}\n

  • Step 2: Expand the numerator using FOIL (First, Outer, Inner, Last):

  • First: $5 \times 5 = 25$

  • Outer: $5 \times i = 5i$

  • Inner: $i \times 5 = 5i$

  • Last: $i \times i = i^2$

  • Numerator total: $25 + 5i + 5i + i^2 = 25 + 10i + i^2$

  • Step 3: Expand the denominator using conjugate shortcut:

  • For conjugates $(a - bi)(a + bi) = a^2 - b^2i^2 = a^2 + b^2$. Outer and inner FOIL terms cancel completely.

  • Denominator total: $5 \times 5 - i \times i = 25 - i^2$

  • Step 4: Substitute the imaginary unit identity $i^2 = -1$:

  • Numerator: $25 + 10i + (-1) = 24 + 10i$

  • Denominator: $25 - (-1) = 25 + 1 = 26$

  • Step 5: Separate terms into $a + bi$ format and reduce fractions:

  • \n\frac{24 + 10i}{26} = \frac{24}{26} + \frac{10i}{26}\n

  • Reduce numerators and denominators by dividing by $2$:

  • Final Answer: \n\frac{12}{13} + \frac{5}{13}i\n

Graphing Calculator Verification Procedures

• TI-84 / TI-SmartView Complex Mode Configuration:

  • Press the MODE key on the calculator (located adjacent to the SECOND key).

  • Scroll down to the row containing REAL mode options.

  • Select a+bi mode and press ENTER to activate complex number functionality.

  • Exit the mode menu using Quit (SECOND + MODE).

• Syntax Rules for Entering Complex Division:

  • Enclose two-term numerators and denominators in explicit parentheses: (5 + i) / (5 - i).

  • Input the imaginary unit $i$ using the key sequence SECOND + . (period/decimal point key).

• Converting Decimal Outputs to Exact Simplified Fractions:

  • Complex divisions often produce decimal outputs on screen.

  • Convert decimal values to standard fraction form by hitting MATH key, selecting option 1: >Frac, and hitting ENTER.

  • Output will display simplified exact fractional values: \n\frac{12}{13} + \frac{5}{13}i\n

• Policy on Assessment Work:

  • Calculator checks serve to verify answers, but complete manual mathematical steps must be recorded to receive full academic credit.

Radical Expressions and Higher Operations

• Quotient Rule for Radicals:

  • Theorem: \n\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\n

  • Problem Example: Simplifying radical quotients: \n\frac{\sqrt{-315}}{\sqrt{45}}\n

  • Efficient Method: Divide radicands directly before simplifying radical components:

  • \n\frac{-315}{45} = -7\n

  • Resulting radical: \n\sqrt{-7}\n

  • Simplify negative sign to imaginary unit $i$: $i\sqrt{7}$.

• Multiplication of Imaginary Radical Expressions:

  • Problem Example: \n\sqrt{-32} \times \sqrt{-17}\n

  • Critical Rule / Pitfall Warning: Negative radicands MUST be simplified into imaginary $i$ terms BEFORE performing multiplication. Multiplying negative numbers directly under radicals ($\sqrt{-32 \times -17} = \sqrt{544}$) cancels the negative signs illegally, producing an incorrect real number and losing imaginary components.

  • Step 1: Simplify each radical individually:

  • \n\sqrt{-32} = \sqrt{-1 \times 16 \times 2} = 4i\sqrt{2}\n

  • \n\sqrt{-17} = i\sqrt{17}\n

  • Step 2: Multiply external terms and internal radicands separately:

  • External terms: $(4i)(i) = 4i^2$

  • Internal radicands: \n\sqrt{2} \times \sqrt{17} = \sqrt{34}\n

  • Expression: $4i^2\sqrt{34}$

  • Step 3: Substitute $i^2 = -1$:

  • $4(-1)\sqrt{34} = -4\sqrt{34}$

  • Final Answer: $-4\sqrt{34}$.

• Squaring a Complex Binomial:

  • Problem Example: $(7 - 6i)^2$

  • Definition of Square: Rewrite expression as binomial multiplication: $(7 - 6i)(7 - 6i)$.

  • Step 1: Expand using FOIL:

  • First: $7 \times 7 = 49$

  • Outer: $7 \times (-6i) = -42i$

  • Inner: $-6i \times 7 = -42i$

  • Last: $(-6i) \times (-6i) = +36i^2$

  • Step 2: Combine like imaginary terms:

  • $-42i - 42i = -84i$

  • Expression: $49 - 84i + 36i^2$

  • Step 3: Convert $i^2$ term:

  • $+36i^2 = 36(-1) = -36$

  • Step 4: Combine real constants:

  • $49 - 36 = 13$

  • Final Solution: $13 - 84i$

• Number System Classifications:

  • Real Numbers: Numbers with no imaginary components ($b = 0$ in $a + bi$).

  • Pure Imaginary Numbers: Numbers with no real part ($a = 0$ and $b \neq 0$ in $a + bi$).

  • Nonreal Complex Numbers: Numbers containing both real and imaginary components ($a \neq 0$ and $b \neq 0$).

Factoring Methods and Zero Factor Property

• Direct Use of Zero Factor Property:

  • Definition: An equation is set up for direct use of the zero factor property if it is completely factored and set equal to zero.

  • Example: $(5x - 1)(x - 5) = 0$

  • Procedure:

  • Set first linear factor to zero: $5x - 1 = 0 \implies 5x = 1 \implies x = \frac{1}{5}$

  • Set second linear factor to zero: $x - 5 = 0 \implies x = 5$

  • Solution set: \n\left\{\frac{1}{5}, 5\right\}\n

• Factoring Trinomials by Grouping (AC Method):

  • Problem Example: $3x^2 - 13x + 10 = 0$

  • Coefficient identification: $a = 3$, $b = -13$, $c = 10$

  • Step 1: Calculate $a \times c$ product:

  • $a \times c = 3 \times 10 = 30$

  • Step 2: Identify factors of $30$ that add up to $b = -13$:

  • Testing factor pairs: $-3$ and $-10$

  • Check: $(-3) \times (-10) = 30$ and $(-3) + (-10) = -13$

  • Step 3: Split the middle linear term $-13x$ into two separate terms:

  • $3x^2 - 3x - 10x + 10 = 0$

  • Step 4: Factor by grouping in pairs:

  • Group 1 ($3x^2 - 3x$): Factor out Greatest Common Factor $3x \implies 3x(x - 1)$

  • Group 2 ($-10x + 10$): Factor out Greatest Common Factor $-10 \implies -10(x - 1)$

  • Combined expression: $3x(x - 1) - 10(x - 1) = 0$

  • Step 5: Extract common binomial factor $(x - 1)$:

  • $(x - 1)(3x - 10) = 0$

  • Step 6: Solve each factor using zero factor property:

  • $x - 1 = 0 \implies x = 1$

  • $3x - 10 = 0 \implies 3x = 10 \implies x = \frac{10}{3}$

• Factoring using Slide and Divide Method:

  • Problem Example: $5c^2 - 26c = 24$

  • Step 1: Set equation equal to zero:

  • $5c^2 - 26c - 24 = 0$

  • Step 2: Multiply $a \times c$:

  • $5 \times (-24) = -120$

  • Step 3: "Slide" leading coefficient $a = 5$ to constant term, creating a monic trinomial:

  • $c^2 - 26c - 120 = 0$

  • Step 4: Factor monic trinomial by finding factor pairs of $-120$ that sum to $-26$:

  • Testing factor pairs: $+4$ and $-30$ ($4 \times (-30) = -120$ and $4 + (-30) = -26$)

  • Monic factored expression: $(c + 4)(c - 30) = 0$

  • Step 5: "Divide" numeric constants inside binomials by original leading coefficient $a = 5$:

  • \n\left(c + \frac{4}{5}\right)\left(c - \frac{30}{5}\right) = 0\n

  • Step 6: Simplify fractions and slide denominators:

  • \n\frac{30}{5} = 6 \implies (c - 6)\n

  • \n\frac{4}{5}\n cannot reduce; slide denominator $5$ in front of variable c \implies (5c + 4)$\n - Complete factored form: (5c + 4)(c - 6) = 0\n - Step 7: Solve linear factors:\n - 5c + 4 = 0 \implies 5c = -4 \implies c = -\frac{4}{5}\n - c - 6 = 0 \implies c = 6\n\n- Grading Policy Note:\n - If assessment instructions state "solve by factoring", using quadratic formula instead of factoring will result in partial credit deduction (e.g., maximum half credit awarded).\n\n# Square Root Property and Literal Equations\n\n- Square Root Property Definition:\n - If u^2 = d$, then$u = \pm\sqrt{d}.\n - Requires variable or binomial square to be isolated on one side of equation.\n\n- Basic Square Root Property Application:\n - Problem Example: x^2 = -25\n - Apply property: x = \pm\sqrt{-25}\n - Simplify imaginary radical: x = \pm 5i\n\n- Rearranging Terms Before Applying Property:\n - Problem Example: 45 - t^2 = 0\n - Step 1: Isolate t^2 term by moving it to right side to maintain positive coefficient:\n - 45 = t^2\n - Step 2: Apply square root property:\n - t = \pm\sqrt{45}\n - Step 3: Simplify radical (\sqrt{45} = \sqrt{9 \times 5}):\n - t = \pm 3\sqrt{5}\n\n- Binomial Square Applications:\n - Problem Example: (3x + 4)^2 = 2\n - Step 1: Apply square root property directly to squared binomial:\n - 3x + 4 = \pm\sqrt{2}\n - Step 2: Subtract 4 from both sides:\n - 3x = -4 \pm \sqrt{2}\n - Step 3: Divide entire expression by 3:\n -  x = \frac{-4 \pm \sqrt{2}}{3} \n\n- Solving Literal Equations using Square Root Property:\n - Problem Example: Solve S = \frac{1}{11}gt^2$for variable$t.\n - Step 1: Clear denominator by multiplying both sides by 11:\n - 11S = gt^2\n - Step 2: Isolate t^2$by dividing both sides by$g:\n - t^2 = \frac{11S}{g} \n - Step 3: Apply square root property to both sides:\n - t = \pm\sqrt{\frac{11S}{g}} = \pm\frac{\sqrt{11S}}{\sqrt{g}} \n - Step 4: Rationalize denominator by multiplying top and bottom by \sqrt{g}:\n - t = \pm\frac{\sqrt{11S} \cdot \sqrt{g}}{\sqrt{g} \cdot \sqrt{g}} = \pm\frac{\sqrt{11Sg}}{g} \n\n# Quadratic Formula Applications\n\n- Standard Form Requirement:\n - Quadratic equations must be written in standard form ax^2 + bx + c = 0$($a \neq 0$) before identifying coefficients$a$,$b$, and$c\n\n- Quadratic Formula Theorem:\n - x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \n\n- Problem Application:\n - Problem Example: r^2 + 3r - 2 = 0\n - Coefficient identification: a = 1$,$b = 3$,$c = -2\n - Step 1: Substitute values into formula:\n - r = \frac{-(3) \pm \sqrt{(3)^2 - 4(1)(-2)}}{2(1)} \n - Step 2: Evaluate discriminant (b^2 - 4ac):\n - 3^2 = 9\n - -4(1)(-2) = +8\n - Discriminant total: 9 + 8 = 17\n - Step 3: Combine components over common denominator:\n - r = \frac{-3 \pm \sqrt{17}}{2} \n - Alternative Two-Fraction Notation:\n - r = -\frac{3}{2} \pm \frac{\sqrt{17}}{2} \n\n# Completing the Square Method\n\n- Procedural Sequence for Completing the Square:\n - Step 1: Ensure constant term c is isolated on right side of equation.\n - Step 2: Ensure leading coefficient a = 1$. If$a \neq 1$, divide every term on both sides by$a$.

  • Step 3: Take linear coefficient $b$, divide it by $2$ ($\frac{b}{2}$), square result ($\left(\frac{b}{2}\right)^2$), and add to both sides.

  • Step 4: Factor left side into perfect square binomial $\left(x + \frac{b}{2}\right)^2$.

  • Step 5: Solve using square root property.

• Problem Application:

  • Problem Example: $-9x^2 + 36x = 41$

  • Step 1: Constant term $41$ is already isolated on right side.

  • Step 2: Divide all terms by leading coefficient $-9$ to make $a = 1$:

  • \n\frac{-9x^2}{-9} + \frac{36x}{-9} = \frac{41}{-9}\n

  • \nx^2 - 4x = -\frac{41}{9}\n

  • Step 3: Calculate square of half the linear coefficient $b = -4$:

  • \n\frac{-4}{2} = -2 \implies (-2)^2 = 4\n

  • Step 4: Add $4$ to both sides of equation:

  • \nx^2 - 4x + 4 = -\frac{41}{9} + 4\n

  • Step 5: Convert integer $4$ to common denominator fraction ($4 = \frac{36}{9}$) and combine right side terms:

  • \n-\frac{41}{9} + \frac{36}{9} = -\frac{5}{9}\n

  • Step 6: Factor left side into perfect square binomial:

  • \n(x - 2)^2 = -\frac{5}{9}\n

  • Step 7: Extract square roots using square root property:

  • \nx - 2 = \pm\sqrt{-\frac{5}{9}} = \pm\frac{i\sqrt{5}}{3}\n

  • Step 8: Add $2$ to both sides to isolate variable $x$:

  • \nx = 2 \pm \frac{i\sqrt{5}}{3}\n

Geometric Application and Area Word Problems

• Uniform Border / Rug Area Problem:

  • Problem Scenario: A rectangular room measures $19\,\text{ft}$ wide by $26\,\text{ft}$ long. A rectangular rug placed in center leaves a floor border of uniform width $x$ surrounding it. The total rug area is $330\,\text{sq ft}$.

  • Variable Definition:

  • Let $x$ = uniform width of exposed floor border surrounding rug (in feet).

  • Rug Dimension Algebraic Modeling:

  • Length of rug = $26 - 2x$

  • Width of rug = $19 - 2x$

  • Area Model Equation:

  • \n\text{Area} = \text{Length} \times \text{Width}\n

  • $(26 - 2x)(19 - 2x) = 330$

  • Expansion and Reduction to Standard Form:

  • FOIL left side: $26(19) - 26(2x) - 2x(19) + (-2x)(-2x) = 330$

  • $494 - 52x - 38x + 4x^2 = 330$

  • Combine linear terms: $4x^2 - 90x + 494 = 330$

  • Subtract $330$ from both sides: $4x^2 - 90x + 164 = 0$

  • Simplify Equation Coefficients:

  • Divide all terms by Greatest Common Divisor $2$:

  • $2x^2 - 45x + 82 = 0$

  • Factoring Quadratic Expression:

  • $(2x - 41)(x - 2) = 0$

  • Solving Linear Factors for Border Width $x$:

  • $2x - 41 = 0 \implies 2x = 41 \implies x = \frac{41}{2} = 20.5\,\text{ft}$

  • $x - 2 = 0 \implies x = 2\,\text{ft}$

  • Physical Constraint Analysis and Selection:

  • $x = 20.5\,\text{ft}$ is physically impossible because total room width is only $19\,\text{ft}$ ($20.5\,\text{ft}$ exceeds room dimensions).

  • Valid floor border width: $x = 2\,\text{ft}$.

  • Final Rug Dimensions Calculation:

  • Length of rug = $26 - 2(2) = 26 - 4 = 22\,\text{ft}$

  • Width of rug = $19 - 2(2) = 19 - 4 = 15\,\text{ft}$

  • Area Verification:

  • \n\text{Area} = 22\,\text{ft} \times 15\,\text{ft} = 330\,\text{sq ft}\n