Quadratic Equations
ICSE · Class 11 · Mathematics
Flashcards for Quadratic Equations — ICSE Class 11 Mathematics. Quick Q&A cards covering key concepts, definitions, and formulas.
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Get startedSolve: x^2 - 5x + 6 = 0
Answer
Step 1: Find two numbers with product 6 and sum -5. Step 2: Write x^2 - 5x + 6 = (x - 2)(x - 3). Step 3: Set each factor equal to zero. (x - 2) = 0 or (x - 3) = 0 Step 4: x = 2 or x = 3 Answer: The ro…
Solve: x^2 + x + 1 = 0 using the quadratic formula
Answer
Step 1: Identify a = 1, b = 1, c = 1. Step 2: Use x = (-b ± √(b^2 - 4ac)) / (2a). Step 3: Substitute values. x = (-1 ± √(1 - 4)) / 2 x = (-1 ± √(-3)) / 2 Step 4: Write √(-3) = i√3. x = (-1 ± i√3) / 2 …
When do you use the quadratic formula? Solve x^2 + 8 = 0 as an example.
Answer
Use the quadratic formula when factoring is difficult or not possible. Step 1: Write the equation in ax^2 + bx + c = 0 form. x^2 + 8 = 0 → a = 1, b = 0, c = 8 Step 2: Apply x = (-b ± √(b^2 - 4ac)) / (…
Find the discriminant of x^2 - 6x + 8 = 0 and state the nature of the roots
Answer
Step 1: Identify a = 1, b = -6, c = 8. Step 2: Use D = b^2 - 4ac. D = (-6)^2 - 4(1)(8) D = 36 - 32 D = 4 Step 3: Since D > 0, the roots are real and distinct. Answer: D = 4, so the roots are real and …
Find the roots of x^2 - 2x - 2 = 0 and classify them
Answer
Step 1: a = 1, b = -2, c = -2. Step 2: Find the discriminant. D = (-2)^2 - 4(1)(-2) = 4 + 8 = 12 Step 3: Since D > 0 and 12 is not a perfect square, the roots are real, irrational, and conjugate. Step…
Find the roots of x^2 - (5 + √2)x + 5√2 = 0
Answer
Step 1: a = 1, b = -(5 + √2), c = 5√2. Step 2: Use the quadratic formula. x = [5 + √2 ± √((5 + √2)^2 - 20√2)] / 2 Step 3: Simplify the discriminant. (5 + √2)^2 = 25 + 10√2 + 2 = 27 + 10√2 27 + 10√2 - …
Solve: x^2 - 2ix - 1 = 0
Answer
Step 1: a = 1, b = -2i, c = -1. Step 2: Use the quadratic formula. x = [2i ± √((-2i)^2 - 4(1)(-1))] / 2 Step 3: Simplify inside the root. (-2i)^2 = -4 -4 + 4 = 0 Step 4: Therefore both roots are equal…
Why do complex roots of a real-coefficient quadratic occur in pairs? Show with an example.
Answer
For a quadratic with real coefficients, if one complex root is a + bi, the other is a - bi. Reason: the non-real part must cancel to keep the coefficients real. Example: x^2 + x + 1 = 0 has roots (-1 …
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