Overview

Substitution turns messy expressions into familiar ones. Look for repeated patterns like $x+1/x$, $x^2$, or symmetric sums.

Key Ideas

• Let $t = x + \frac{1}{x}$ to collapse symmetric expressions.
• Let $u = x^2$ to reduce degree in even powers.
• Use $s=x+y$, $p=xy$ when expressions are symmetric in two variables.

Core Skills

Identify Repeated Structure

Look for the same expression appearing multiple times, then replace it with a single variable to reduce complexity.

Translate Back Carefully

After solving in the new variable, substitute back and check that all solutions fit the original domain.

Pair With Identities

Use identities like $x^2+1/x^2 = t^2-2$ to translate higher powers.

Worked Example

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

Square the equation: $\left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} = 25$. So $x^2 + \frac{1}{x^2} = 25 - 2 = 23$.

More Examples

Example 1: Even Powers

If $x^4-5x^2+4=0$, solve for $x$.

Let $u=x^2$. Then $u^2-5u+4=0$ so $u=1$ or $4$. Hence $x=\pm 1,\pm 2$.

Example 2: Symmetric System

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

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

Example 3: Reciprocal Form

If $x+1/x=3$, find $x^3+1/x^3$.

Use $x^3+1/x^3=(x+1/x)^3-3(x+1/x)=27-9=18$.

Strategy Checklist

• Identify the repeated expression.
• Substitute consistently across all terms.
• Translate back and verify domain restrictions.

Common Pitfalls

• Forgetting domain restrictions, such as $x \ne 0$.
• Substituting but not translating all terms consistently.
• Keeping extraneous solutions after squaring.

Practice Problems