Overview

Algebra in contests rewards clean manipulation: simplify, factor, and solve. Build habits like checking units, substituting smart values, and rewriting in equivalent forms.

Key Ideas

• Solve linear equations by isolating variables; keep track of constraints.
• Factor common patterns: $a^2 - b^2 = (a-b)(a+b)$ and $a^2 + 2ab + b^2$.
• Use substitution to reduce complex systems to a single variable.
• Clear denominators carefully and record excluded values.
• Try to factor before expanding; it often reveals structure.

Core Skills

Linear Equations

Collect like terms, isolate the variable, and verify constraints. For equations with fractions, clear denominators first, then check for extraneous solutions.

Factoring Patterns

Common patterns to recognize:

• Difference of squares: $a^2-b^2=(a-b)(a+b)$.
• Perfect square trinomials: $a^2\pm2ab+b^2=(a\pm b)^2$.
• Quadratic factoring: $x^2+bx+c=(x+p)(x+q)$ with $p+q=b$, $pq=c$.

Substitution

If the same expression appears repeatedly (like $x+1/x$ or $x+y$), introduce a new variable to simplify the system.

Clearing Denominators

Multiply both sides by the least common denominator (LCD) and track values that make the LCD zero. Those values are not allowed.

Worked Example

Solve $\frac{3}{x} + \frac{2}{x+1} = 1$.

Multiply both sides by $x(x+1)$ to clear denominators: $3(x+1) + 2x = x(x+1).$ So $3x + 3 + 2x = x^2 + x$, giving $x^2 - 4x - 3 = 0$. Then $x = 2 \pm \sqrt{7}$. Check that neither solution makes a denominator $0$.

More Examples

Example 1: Factoring

Factor $x^2-9$.

$x^2-9=(x-3)(x+3)$.

Example 2: Substitution

If $x + \frac{1}{x} = 5$, find $x^2 + \frac{1}{x^2}$.

Square both sides: $x^2 + 2 + \frac{1}{x^2} = 25$, so $x^2 + \frac{1}{x^2} = 23$.

Example 3: Linear System via Sum and Product

If $x+y=7$ and $xy=10$, find $x^2+y^2$.

$x^2+y^2=(x+y)^2-2xy=49-20=29$.

Strategy Checklist

• Are there hidden restrictions (like denominators or square roots)?
• Can you factor instead of expand?
• Is there a repeated expression that suggests substitution?
• Can the equation be rewritten into a known identity?

Common Pitfalls

• Clearing denominators but forgetting to track excluded values.
• Expanding when factoring is faster.
• Dropping negative signs when distributing.
• Forgetting to check solutions back in the original equation.

Practice Problems