🔢 Algebra Tactics

Master the essential algebraic techniques that will help you solve equations, manipulate expressions, and handle complex algebraic relationships efficiently.

🎯 Core Algebraic Strategies

Substitution Techniques

• Variable substitution: Replace complex expressions with simpler variables
• Trigonometric substitution: Use trig identities for algebraic problems
• Symmetric substitution: Exploit symmetry in expressions
• Parameter substitution: Introduce parameters to simplify

Discriminant Analysis

• Quadratic discriminant: $b^2 - 4ac$ determines nature of roots
• Positive discriminant: Two distinct real roots
• Zero discriminant: One repeated real root
• Negative discriminant: Two complex roots

Conjugate Techniques

• Rationalizing denominators: Multiply by conjugate
• Complex conjugates: Use $z \cdot \overline{z} = |z|^2$
• Radical conjugates: Simplify expressions with radicals
• Trigonometric conjugates: Use complementary angles

🔄 Substitution Mastery

Variable Substitution

When to use: Complex expressions with repeated patterns

Process:

1. Identify the pattern: Look for repeated expressions
2. Choose substitution: Let $u = \text{pattern}$
3. Rewrite equation: Express in terms of $u$
4. Solve for $u$: Use standard techniques
5. Back-substitute: Find original variable

Example: Solve $x^4 - 5x^2 + 4 = 0$

Solution:

• Let $u = x^2$, then $u^2 - 5u + 4 = 0$
• Factor: $(u-1)(u-4) = 0$, so $u = 1$ or $u = 4$
• Back-substitute: $x^2 = 1$ or $x^2 = 4$
• Solutions: $x = \pm 1, \pm 2$

Symmetric Substitution

When to use: Symmetric expressions in multiple variables

Process:

1. Identify symmetry: Look for symmetric patterns
2. Use symmetric variables: $s = x + y$, $p = xy$
3. Express in terms: Rewrite using symmetric variables
4. Solve systematically: Use symmetric properties
5. Find original variables: Back-substitute

Example: Solve $x + y = 5$, $xy = 6$

Solution:

• Let $s = x + y = 5$, $p = xy = 6$
• $x, y$ are roots of $t^2 - st + p = 0$
• $t^2 - 5t + 6 = 0$, so $(t-2)(t-3) = 0$
• Solutions: $(x,y) = (2,3)$ or $(3,2)$

📊 Discriminant Analysis

Quadratic Discriminant

Formula: $\Delta = b^2 - 4ac$ for $ax^2 + bx + c = 0$

Cases:

• $\Delta > 0$: Two distinct real roots
• $\Delta = 0$: One repeated real root
• $\Delta < 0$: Two complex roots

Applications:

• Nature of roots: Determine if roots are real
• Number of solutions: Count real solutions
• Parameter ranges: Find values for specific root types
• Optimization: Find extrema of quadratic functions

Discriminant Example

Problem: For what values of $k$ does $x^2 + kx + 1 = 0$ have real roots?

Solution:

• Discriminant: $\Delta = k^2 - 4(1)(1) = k^2 - 4$
• Real roots when $\Delta \geq 0$: $k^2 - 4 \geq 0$
• $k^2 \geq 4$, so $|k| \geq 2$
• Answer: $k \leq -2$ or $k \geq 2$

🔗 Conjugate Techniques

Rationalizing Denominators

When to use: Denominators with radicals or complex numbers

Process:

1. Identify conjugate: Change sign of radical or imaginary part
2. Multiply by conjugate: Both numerator and denominator
3. Simplify: Use difference of squares or other identities
4. Check result: Verify the simplification

Example: Rationalize $\frac{1}{\sqrt{3} + \sqrt{2}}$

Solution:

• Conjugate: $\sqrt{3} - \sqrt{2}$
• Multiply: $\frac{1}{\sqrt{3} + \sqrt{2}} \cdot \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}}$
• Simplify: $\frac{\sqrt{3} - \sqrt{2}}{(\sqrt{3})^2 - (\sqrt{2})^2} = \frac{\sqrt{3} - \sqrt{2}}{3 - 2} = \sqrt{3} - \sqrt{2}$

Complex Conjugates

When to use: Complex numbers and expressions

Key identity: $z \cdot \overline{z} = |z|^2$

Applications:

• Modulus calculation: $|z|^2 = z \cdot \overline{z}$
• Division: $\frac{z_1}{z_2} = \frac{z_1 \cdot \overline{z_2}}{|z_2|^2}$
• Real parts: $\text{Re}(z) = \frac{z + \overline{z}}{2}$
• Imaginary parts: $\text{Im}(z) = \frac{z - \overline{z}}{2i}$

🎯 Advanced Algebraic Techniques

Vieta’s Formulas

For quadratic: If $ax^2 + bx + c = 0$ has roots $r_1, r_2$:

• $r_1 + r_2 = -\frac{b}{a}$
• $r_1 \cdot r_2 = \frac{c}{a}$

For cubic: If $ax^3 + bx^2 + cx + d = 0$ has roots $r_1, r_2, r_3$:

• $r_1 + r_2 + r_3 = -\frac{b}{a}$
• $r_1r_2 + r_1r_3 + r_2r_3 = \frac{c}{a}$
• $r_1r_2r_3 = -\frac{d}{a}$

Vieta’s Example

Problem: If $x^2 - 5x + 6 = 0$ has roots $r_1, r_2$, find $r_1^2 + r_2^2$

Solution:

• Vieta’s: $r_1 + r_2 = 5$, $r_1r_2 = 6$
• $r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2r_1r_2 = 5^2 - 2(6) = 25 - 12 = 13$

Polynomial Division

Synthetic division: For dividing by linear factors Long division: For dividing by higher degree polynomials Remainder theorem: $P(a)$ is remainder when $P(x)$ divided by $(x-a)$ Factor theorem: $(x-a)$ is factor of $P(x)$ if and only if $P(a) = 0$

⚡ Quick Algebraic Tricks

Factoring Techniques

• Common factor: Factor out common terms
• Difference of squares: $a^2 - b^2 = (a+b)(a-b)$
• Perfect squares: $a^2 \pm 2ab + b^2 = (a \pm b)^2$
• Sum/difference of cubes: $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$
• Grouping: Group terms to factor

Completing the Square

Process:

1. Isolate variable terms: Move constant to other side
2. Factor coefficient: Factor out coefficient of $x^2$
3. Add and subtract: Add $(\frac{b}{2})^2$ inside parentheses
4. Simplify: Write as perfect square
5. Solve: Take square root and solve

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

Solution:

• $x^2 - 6x = -5$
• $x^2 - 6x + 9 = -5 + 9$ (add $(\frac{-6}{2})^2 = 9$)
• $(x-3)^2 = 4$
• $x-3 = \pm 2$, so $x = 3 \pm 2 = 1, 5$

🎯 Problem-Specific Strategies

Equation Solving

• Linear equations: Isolate variable
• Quadratic equations: Factor, complete square, or use formula
• Rational equations: Clear denominators
• Radical equations: Isolate radical, square both sides
• Exponential equations: Use logarithms
• Logarithmic equations: Use exponential form

Function Analysis

• Domain: Values where function is defined
• Range: Possible output values
• Intercepts: Where function crosses axes
• Symmetry: Even, odd, or neither
• Asymptotes: Vertical, horizontal, or oblique
• Extrema: Maximum and minimum values

Inequality Solving

• Linear inequalities: Solve like equations, flip sign when multiplying by negative
• Quadratic inequalities: Find roots, test intervals
• Rational inequalities: Find zeros and undefined points, test intervals
• Absolute value inequalities: Use definition of absolute value
• Compound inequalities: Solve each part separately

🚨 Common Algebraic Mistakes

Avoid These Errors:

• Sign errors: Be careful with negative signs
• Distribution errors: Don’t forget to distribute
• Factoring errors: Check your factoring
• Domain errors: Don’t divide by zero
• Extraneous solutions: Check solutions in original equation

Red Flags:

• Answer doesn’t work: Check your work
• Negative under square root: Check your work
• Division by zero: Check your work
• Impossible result: Check your work

📊 Quick Reference

Substitution Checklist:

• Identify pattern: Look for repeated expressions
• Choose substitution: Let $u = \text{pattern}$
• Rewrite equation: Express in terms of $u$
• Solve for $u$: Use standard techniques
• Back-substitute: Find original variable

Discriminant Checklist:

• Identify quadratic: $ax^2 + bx + c = 0$
• Calculate discriminant: $\Delta = b^2 - 4ac$
• Determine nature: Positive, zero, or negative
• Apply result: Use discriminant information

Conjugate Checklist:

• Identify conjugate: Change sign of radical or imaginary part
• Multiply by conjugate: Both numerator and denominator
• Simplify: Use difference of squares or other identities
• Check result: Verify the simplification

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