Factor Flip WarmupFactor Flip Warmup Quadratic Equations • Factoring Name: Date: Period: Example \( x^2 - 7x + 12 = 0 \) Factored \( (x - 3)(x - 4) = 0 \) → Zero Product \( x - 3 = 0 \quad \text{or} \quad x - 4 = 0 \) Solutions \( x = 3, \quad x = 4 \) 1 Problem 1 Solutions: \( x = -5, \quad x = 2 \) Work Standard Form = 0 2 Problem 2 Solutions: \( x = \frac{1}{2}, \quad x = -\frac{2}{3} \) Work Standard Form = 0
Factor Flip Answer KeyFactor Flip Answer Key Teacher Reference • Solutions & Grading Guidance Answer Key Example Model \( x^2 - 7x + 12 = 0 \) \( (x - 3)(x - 4) = 0 \) → \( x = 3, \quad x = 4 \) 1 Problem 1 Solutions Given: \( x = -5, \quad x = 2 \) Step-by-Step Work 1. Linear factors: \( (x - (-5)) \implies (x + 5) \), \( (x - 2) \) 2. Factored form: \( (x + 5)(x - 2) = 0 \) 3. Expand: \( x^2 - 2x + 5x - 10 = 0 \) 4. Combine: \( x^2 + 3x - 10 = 0 \) Standard Form \( x^2 + 3x - 10 = 0 \) \( a = 1, \; b = 3, \; c = -10 \) Watch for sign errors: Students frequently write \( (x - 5)(x + 2) = 0 \), resulting in \( x^2 - 3x - 10 = 0 \). Remind them that if \( x = -5 \), then \( x + 5 = 0 \). 2 Problem 2 Solutions Given: \( x = \frac{1}{2}, \quad x = -\frac{2}{3} \) Expected Methods Method A: Integer Factors (Clearing Fractions) \( x = \frac{1}{2} \implies 2x - 1 = 0 \) \( x = -\frac{2}{3} \implies 3x + 2 = 0 \) \( (2x - 1)(3x + 2) = 6x^2 + 4x - 3x - 2 = 0 \) Method B: Direct Monic Factors \( (x - \frac{1}{2})(x + \frac{2}{3}) = x^2 + \frac{1}{6}x - \frac{1}{3} = 0 \) Standard Form (Integer Coefficients Preferred) \( 6x^2 + x - 2 = 0 \) Also accept: \( x^2 + \frac{1}{6}x - \frac{1}{3} = 0 \) Teaching Tip: Highlight that both forms describe the exact same parabola scaled by a factor of 6, both sharing the exact same \(x\)-intercepts. Recommended Opener Pacing: 4 min independent work • 2 min student check • 2 min whole-class debrief Key: #1 \(x^2+3x-10=0\), #2 \(6x^2+x-2=0\)
Root Reversal Practice WorksheetRoot Reversal Practice Worksheet Quadratic Equations • Standard Form \( ax^2 + bx + c = 0 \) Name: Date: Period: 1 Problem 1 Solutions: \( x = 2, \quad x = 5 \) Work Standard Form: = 0 2 Problem 2 Solutions: \( x = -3, \quad x = 7 \) Work Standard Form: = 0 3 Problem 3 Solutions: \( x = -4, \quad x = -8 \) Work Standard Form: = 0 4 Problem 4 Solutions: \( x = 6, \quad x = -6 \) Work Standard Form: = 0 5 Problem 5 Solutions: \( x = 3, \quad x = -\frac{1}{4} \) Work Standard Form: = 0 6 Problem 6 Solutions: \( x = \frac{2}{3}, \quad x = -\frac{3}{2} \) Work Standard Form: = 0 Standard Form: \( ax^2 + bx + c = 0 \) Integer coefficients preferred
Root Reversal Answer KeyRoot Reversal Answer Key Teacher Reference • Complete Worked Solutions Answer Key 1 Problem 1 Given: \( x = 2, \quad x = 5 \) Work \( (x - 2)(x - 5) = 0 \) \( x^2 - 5x - 2x + 10 = 0 \) Combine like linear terms Standard Form: \( x^2 - 7x + 10 = 0 \) 2 Problem 2 Given: \( x = -3, \quad x = 7 \) Work \( (x + 3)(x - 7) = 0 \) \( x^2 - 7x + 3x - 21 = 0 \) Note sign change: \( x - (-3) = x + 3 \) Standard Form: \( x^2 - 4x - 21 = 0 \) 3 Problem 3 Given: \( x = -4, \quad x = -8 \) Work \( (x + 4)(x + 8) = 0 \) \( x^2 + 8x + 4x + 32 = 0 \) Both roots negative → all positive terms Standard Form: \( x^2 + 12x + 32 = 0 \) 4 Problem 4 Given: \( x = 6, \quad x = -6 \) Work \( (x - 6)(x + 6) = 0 \) \( x^2 + 6x - 6x - 36 = 0 \) Difference of squares (\(b = 0\)) Standard Form: \( x^2 - 36 = 0 \) 5 Problem 5 Given: \( x = 3, \quad x = -\frac{1}{4} \) Work \( (x - 3)(4x + 1) = 0 \) \( 4x^2 + x - 12x - 3 = 0 \) Also accept monic: \( x^2 - \frac{11}{4}x - \frac{3}{4} = 0 \) Standard Form: \( 4x^2 - 11x - 3 = 0 \) 6 Problem 6 Given: \( x = \frac{2}{3}, \quad x = -\frac{3}{2} \) Work \( (3x - 2)(2x + 3) = 0 \) \( 6x^2 + 9x - 4x - 6 = 0 \) Also accept monic: \( x^2 + \frac{5}{6}x - 1 = 0 \) Standard Form: \( 6x^2 + 5x - 6 = 0 \) 1: \(x^2-7x+10=0\) 2: \(x^2-4x-21=0\) 3: \(x^2+12x+32=0\) 4: \(x^2-36=0\) 5: \(4x^2-11x-3=0\) 6: \(6x^2+5x-6=0\)
Root Reversal Mastery WorksheetRoot Reversal Mastery Worksheet Quadratic Equations • Special Cases & Rational Roots Name: Date: Period: 1 Problem 1 Solutions: \( x = 0, \quad x = -5 \) Work Standard Form: = 0 2 Problem 2 Solutions: \( x = 4 \text{ (double root)} \) Work Standard Form: = 0 3 Problem 3 Solutions: \( x = -9, \quad x = 2 \) Work Standard Form: = 0 4 Problem 4 Solutions: \( x = \frac{5}{2}, \quad x = -\frac{5}{2} \) Work Standard Form: = 0 5 Problem 5 Solutions: \( x = -\frac{2}{5}, \quad x = 3 \) Work Standard Form: = 0 6 Problem 6 Solutions: \( x = -\frac{3}{4}, \quad x = \frac{1}{2} \) Work Standard Form: = 0 Standard Form: \( ax^2 + bx + c = 0 \) Integer coefficients preferred
Root Reversal Mastery Answer KeyRoot Reversal Mastery Answer Key Teacher Reference • Complete Worked Solutions Answer Key 1 Problem 1 Given: \( x = 0, \quad x = -5 \) Work \( x(x - (-5)) = 0 \) \( x(x + 5) = 0 \) \( x^2 + 5x = 0 \) Zero root means constant term \( c = 0 \) Standard Form: \( x^2 + 5x = 0 \) 2 Problem 2 Given: \( x = 4 \text{ (double root)} \) Work \( (x - 4)^2 = (x - 4)(x - 4) = 0 \) \( x^2 - 4x - 4x + 16 = 0 \) Perfect square trinomial; single vertex on x-axis Standard Form: \( x^2 - 8x + 16 = 0 \) 3 Problem 3 Given: \( x = -9, \quad x = 2 \) Work \( (x + 9)(x - 2) = 0 \) \( x^2 - 2x + 9x - 18 = 0 \) Check middle sign: \( +9 - 2 = +7 \) Standard Form: \( x^2 + 7x - 18 = 0 \) 4 Problem 4 Given: \( x = \frac{5}{2}, \quad x = -\frac{5}{2} \) Work \( (2x - 5)(2x + 5) = 0 \) \( 4x^2 + 10x - 10x - 25 = 0 \) Difference of squares (\(b = 0\)); monic: \( x^2 - \frac{25}{4} = 0 \) Standard Form: \( 4x^2 - 25 = 0 \) 5 Problem 5 Given: \( x = -\frac{2}{5}, \quad x = 3 \) Work \( (5x + 2)(x - 3) = 0 \) \( 5x^2 - 15x + 2x - 6 = 0 \) Monic alternative: \( x^2 - \frac{13}{5}x - \frac{6}{5} = 0 \) Standard Form: \( 5x^2 - 13x - 6 = 0 \) 6 Problem 6 Given: \( x = -\frac{3}{4}, \quad x = \frac{1}{2} \) Work \( (4x + 3)(2x - 1) = 0 \) \( 8x^2 - 4x + 6x - 3 = 0 \) Monic alternative: \( x^2 + \frac{1}{4}x - \frac{3}{8} = 0 \) Standard Form: \( 8x^2 + 2x - 3 = 0 \) 1: \(x^2+5x=0\) 2: \(x^2-8x+16=0\) 3: \(x^2+7x-18=0\) 4: \(4x^2-25=0\) 5: \(5x^2-13x-6=0\) 6: \(8x^2+2x-3=0\)
