Theory of Equations

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Using Vieta's formula for quadratic ax² + bx + c = 0: α + β = -b/a = -(-7)/1 = 7 αβ = c/a = 12/1 = 12 Answer: Sum = 7, Product = 12 Note: This direct relationship between roots and coefficients save…

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Using formula: x² - (sum of roots)x + (product of roots) = 0 Step 1: Sum of roots = 5 + (-3) = 2 Step 2: Product of roots = 5 × (-3) = -15 Step 3: Equation: x² - 2x + (-15) = 0 Answer: x² - 2x - 15 …

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Step 1: Find α + β and αβ from original equation α + β = 7/2 αβ = 13/2 Step 2: Find sum of new roots α² + β² = (α + β)² - 2αβ = (7/2)² - 2(13/2) = 49/4 - 26/2 = 49/4 - 52/4 = -3/4 Step 3: Find produ…

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Use Rational Root Theorem when: - Polynomial has integer coefficients - You want to guess possible rational roots - Factoring by inspection is difficult For ax^n + ... + a₀ = 0, possible rational roo…

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This equation has only even powers. Use substitution y = x² Step 1: Substitute to get quadratic y² - 9y + 20 = 0 Step 2: Factor or use quadratic formula (y - 4)(y - 5) = 0 y = 4 or y = 5 Step 3: Su…

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Step 1: Check if sum of coefficients = 0 Sum = 1 + (-3) + (-33) + 35 = 0 Therefore, x = 1 is a root (P(1) = 0) Step 2: Divide polynomial by (x - 1) x³ - 3x² - 33x + 35 = (x - 1)(x² - 2x - 35) Step 3…

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Step 1: Check if sum of coefficients of odd and even powers are equal Odd powers: 11 + (-9) = 2 Even powers: 2 + (-18) = -16 Not equal, so check P(-1): P(-1) = -2 + 11 + 9 - 18 = 0 Therefore, x = -1 i…

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Let α, β, γ, δ be the roots. From ax⁴ + bx³ + cx² + dx + e = 0: Step 1: Find Vieta's relations α + β + γ + δ = -b/a = -2/1 = -2 αβ + αγ + αδ + βγ + βδ + γδ = c/a = -7/1 = -7 Step 2: Use algebraic id…