Quadratic Equation Questions

• Identify the quadratic equation: Which of the following is a quadratic equation (an equation of degree 2)?
• Which equation has real roots? Among the following equations, identify the one that definitely has real roots.
• Solve the quadratic completely: Find the roots of 2x^2 − 9x − 18 = 0.
• Parameter from a known root: If one root of x^2 − 6kx + 5 = 0 is 5, find the value of k.
• Find the roots exactly: Solve 2x^2 − 11x + 15 = 0.
• Evaluate a^3 + 1/a^3 from a squared sum: If (a + 1/a)^2 = 3, find the value of a^3 + 1/a^3.
• Homogeneous quadratic in x and y (ratio form): If 2x^2 − 7xy + 3y^2 = 0 for nonzero x and y, determine the possible values of the ratio x : y.
• Condition for reciprocal roots (standard form): If the quadratic equation (x^2)/a + (x)/b + 1/c = 0 has one root equal to the reciprocal of the other (both nonzero), which condition among a, b, c must hold?
• Find the other root (Vieta's use): One root of the quadratic equation x^2 − 5x + 6 = 0 is 3. Determine the other root.
• Construct a quadratic from sum and product of roots: Form the quadratic equation whose roots have sum 6 and product −16.
• Compare x and y (define the answer codes): I. 4x + 7y = 209 II. 12x − 14y = −38 Choose one option: A) x > y B) x < y C) x = y D) Relationship cannot be determined
• Compare x and y (define the answer codes): I. 18x^2 + 18x + 4 = 0 II. 12y^2 + 29y + 14 = 0 x and y are real roots (any root from each). Choose: A) x > y B) x < y C) x = y D) Relationship cannot be determined
• Compare x and y (define the answer codes): I. 8x^2 + 6x = 5 II. 12y^2 − 22y + 8 = 0 x and y are real roots (any root from each). Choose: A) x > y B) x < y C) x = y D) Relationship cannot be determined
• Compare x and y (define the answer codes): I. 16x^2 + 20x + 6 = 0 II. 10y^2 + 38y + 24 = 0 x and y are real roots (any root from each). Choose: A) x > y B) x < y C) x = y D) Relationship cannot be determined
• Compare x and y (define the answer codes): I. 17x^2 + 48x = 9 II. 13y^2 = 32y − 12 Choose: A) x > y B) x < y C) x = y D) Relationship cannot be determined
• Roots of a parameterized quadratic: Find the roots of a^2 x^2 − 3ab x + 2b^2 = 0 in terms of a and b (assume a ≠ 0).
• Compute α^3 + β^3 from ax^2 + bx + c = 0: If α and β are roots of ax^2 + bx + c = 0 (a ≠ 0), find α^3 + β^3 in terms of a, b, c.
• Real factorization condition: For which values of k does the quadratic polynomial 3z^2 + 5z + k factor over the reals (i.e., has real linear factors)?
• Equal roots condition: The quadratic x^2 + px + q = 0 has equal roots if and only if:
• Consecutive positive odd integers (sum of squares = 290): Find two consecutive positive odd integers whose squares add up to 290.