A quadratic equation x²+bx+c=0 has two real roots. If the difference between the reciprocals of the roots is 1/3, and the sum of the reciprocals of the squares of the roots is 5/9, then the largest possible value of (b+c) is:
Question:
A quadratic equation x²+bx+c=0 has two real roots. If the difference between the reciprocals of the roots is 1/3, and the sum of the reciprocals of the squares of the roots is 5/9, then the largest possible value of (b+c) is:
A quadratic equation x²+bx+c=0 has two real roots. If the difference between the reciprocals of the roots is 1/3, and the sum of the reciprocals of the squares of the roots is 5/9, then the largest possible value of (b+c) is:
Options
Answer: 9
Explanation:
Let the roots of x² + b x + c = 0 be α and β. Step 1: Express conditions 1. Difference of reciprocals: 1/α − 1/β = 1/3 → (β − α)/(α β) = 1/3 → β − α = c/3 (since α β = c) 2. Sum of reciprocals of squares: 1/α² + 1/β² = 5/9 → (α² + β²)/(α β)² = 5/9 → α² + β² = (5/9) c² Step 2: Express α² + β² in terms of b and c α² + β² = (α + β)² − 2 α β = b² − 2 c So: b² − 2 c = (5/9) c² → 9 b² − 18 c = 5 c² → 5 c² + 18 c − 9 b² = 0 Step 3: Express (β − α)² (β − α)² = (α + β)² − 4 α β = b² − 4 c Also, β − α = c/3 → (β − α)² = c² / 9 So: b² − 4 c = c² / 9 → 9 b² − 36 c = c² → c² + 36 c − 9 b² = 0 Step 4: Solve system Equations: 1. 5 c² + 18 c − 9 b² = 0 2. c² + 36 c − 9 b² = 0 Subtract: 4 c² − 18 c = 0 → 2 c (2 c − 9) = 0 → c = 0 or c = 9/2 Step 5: Corresponding b values * If c = 0 → 0 + 36*0 − 9 b² = 0 → b = 0 → b + c = 0 * If c = 9/2 → 81/4 + 162 − 9 b² = 0 → 9 b² = 729/4 → b = ±9/2 → b + c = 9 or 0 Step 6: Largest possible value of b + c Answer: 9
Explanation:
Let the roots of x² + b x + c = 0 be α and β. Step 1: Express conditions 1. Difference of reciprocals: 1/α − 1/β = 1/3 → (β − α)/(α β) = 1/3 → β − α = c/3 (since α β = c) 2. Sum of reciprocals of squares: 1/α² + 1/β² = 5/9 → (α² + β²)/(α β)² = 5/9 → α² + β² = (5/9) c² Step 2: Express α² + β² in terms of b and c α² + β² = (α + β)² − 2 α β = b² − 2 c So: b² − 2 c = (5/9) c² → 9 b² − 18 c = 5 c² → 5 c² + 18 c − 9 b² = 0 Step 3: Express (β − α)² (β − α)² = (α + β)² − 4 α β = b² − 4 c Also, β − α = c/3 → (β − α)² = c² / 9 So: b² − 4 c = c² / 9 → 9 b² − 36 c = c² → c² + 36 c − 9 b² = 0 Step 4: Solve system Equations: 1. 5 c² + 18 c − 9 b² = 0 2. c² + 36 c − 9 b² = 0 Subtract: 4 c² − 18 c = 0 → 2 c (2 c − 9) = 0 → c = 0 or c = 9/2 Step 5: Corresponding b values * If c = 0 → 0 + 36*0 − 9 b² = 0 → b = 0 → b + c = 0 * If c = 9/2 → 81/4 + 162 − 9 b² = 0 → 9 b² = 729/4 → b = ±9/2 → b + c = 9 or 0 Step 6: Largest possible value of b + c Answer: 9
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