Hidden Quadratic Forms

Unit: Analytical Exponential Methods

Mastering U-Substitution in Transcendental Equations

Can You See the Pattern?

\[ e^{2x} - 5e^x + 6 = 0 \]

This doesn't look like a standard linear exponential equation. But what if we looked at it through a different lens?

Observation A

The first term is the square of the second term: \( (e^x)^2 = e^{2x} \).

Observation B

The structure matches \( ax^2 + bx + c = 0 \).

The U-Substitution Workflow

01

Define

Identify the repeated exponential term. Set \( u = e^x \) (or \( u = b^x \)).

02

Transform

Rewrite the equation as a quadratic in terms of \( u \). Solve for \( u \) using factoring or the quadratic formula.

03

Back-Substitute

Replace \( u \) with the original exponential. Solve the resulting simple exponential equations for \( x \).

Worked Example

Equation:

\[ 2^{2x} - 12(2^x) + 32 = 0 \]

Step 1: Let \( u = 2^x \)

\[ u^2 - 12u + 32 = 0 \]

Step 2: Factor

\[ (u - 8)(u - 4) = 0 \]

\[ u = 8, \quad u = 4 \]

Back-Substitution

\( 2^x = 8 \implies x = 3 \)

\( 2^x = 4 \implies x = 2 \)

Critical Note:

If \( u \) is negative, there is no real solution for that branch (e.g., \( e^x = -2 \)).

Workshop Challenge

10 Minutes

Problem 01

\[ 3^{2x} - 10(3^x) + 9 = 0 \]

Problem 02

\[ e^{2x} + 2e^x - 15 = 0 \]

Strategy Check

Identify the \( u \)-substitution base.
Watch for "Extraneous" solutions where \( u \leq 0 \).
Verify your results by substituting back.

Analytical Exponential Methods

Hidden Quadratic Forms

Distinct Base Dilemmas

Exponential Inequality Frontiers

Nonlinear Systems Intersection

Beyond Algebraic Limits

Quadratic Exponential Slides

Hidden Polynomials Worksheet

Hidden Polynomials Answer Key

Distinct Base Slides

Log Distribution Workshop

Log Distribution Answer Key

Exponential Inequality Slides

Interval Mastery Practice

Interval Mastery Answer Key

Exponential Systems Slides

Collision Course Activity

Collision Course Answer Key

Transcendental Limits Slides

Numerical Methods Lab

Numerical Methods Answer Key

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