Formula Blueprint

11th Grade Honors Math: Non-Monic Quadratic Analysis

Architect

Date

Phase 1: Foundation Prep

To complete the square when \(a > 1\), we must first factor out the leading coefficient from the variable terms. Practice creating fractions from integers.

1. Factor out 2:

\(2x^2 - 9x = 2( \quad \quad \quad \quad \quad \quad )\)

2. Factor out 3:

\(3x^2 + 5x = 3( \quad \quad \quad \quad \quad \quad )\)

Phase 2: Site Observation

Video: How To Solve Quadratic Equations By Completing The Square
Target Segment: 7:29 to End (Example 3: \(2x^2 - 7x + 3 = 0\))

1

Pause at 8:30: Discussion Point

Why can't we simply divide the linear coefficient (-7) by 2 and square it immediately?

2

Pause at 9:50: The Multiplier Challenge

The narrator adds \( (\frac{7}{4})^2 \) inside the parentheses on the left. Predict why the right side of the equation must add \( 2 \cdot (\frac{7}{4})^2 \) instead of just \( (\frac{7}{4})^2 \).

Phase 3: The Master Plan (Derivation)

Apply the method observed in the video to the general form equation: \(ax^2 + bx + c = 0\)

Instructional Step	Your Mathematical Work
1. Move the Constant Isolate variable terms on the left.
2. Factor out \(a\) Factor the leading coefficient from variable terms.
3. Complete the Square Find half of the linear term coefficient, square it, and add to both sides (remember the multiplier!).
4. Factor & Simplify RHS Factor the LHS into a binomial squared. Find a common denominator for the RHS.
5. Isolate & Solve for \(x\) Divide by \(a\), take the square root, and solve for \(x\).

Structural Inspection

DISCRIMINANT ORIGIN

In your derivation, at what step did the expression \(b^2 - 4ac\) first appear? Why is this specific value so critical to the solution's existence?

THE \(\pm\) SIGN

Explain the geometric or algebraic necessity of the plus-minus sign when moving from step 4 to step 5.

The Quadratic Blueprint

Formula Blueprint Worksheet

Square Solver Transcript

Similar Lessons

Vector Voyages

Erin Esswein

Love Stakes Game Show

Kelly Brown

Data Detectives

egambale

Data Detectives

Erin Esswein

Data Detective Lab

Erin Esswein

Math Mastery

Katie Heck

Quadratic Factor Quest

Gregory Jordan

Blueprint Builders Polynomial Addition

Cheydon Krysl

Wave Navigator

Abha Poddar

Function Foundations

Jelani Arnold

Parabola Talk

Quan Nguyen

Linear Language

Grant Johnston

Polynomial Puzzle Factoring

Cheydon Krysl

Polynomial Power

Keith Owens

TSIA2 Target Practice

Jessica Waters