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Quadratic Equation |. Standard Form & Basics Standard Form: ax? + bx + c = 0, where a #0 Roots/Solutions: Values of 'x' that satisfy the equation. Degree: 2 (Quadratic) => Two roots possible (Real or Imaginary). ll. Finding Roots - A. Factoring (Splitting the Middle Term): Find two numbers whose sum is ‘b' and product is ‘ac’. Rewrite the middle term (bx) using these two numbers. Factor by grouping. Set each factor to zero and solve for x. Trick: Works best when roots are integers or simple fractions. B. Quadratic Formula: x = (-b + V(b? - 4ac)) / 2a Trick: Use when factoring is difficult or impossible (irrational roots). C. Completing the Square: Useful for deriving the quadratic formula and in specific situations. Steps: Divide equation by "a" if a !=1 Move the constant term (c/a) to the right side. Add (b/2a)? to both sides to complete the square. Rewrite the left side as a squared term. Take the square root of both sides and solve for x. lll. Nature of Roots - The Discriminant (A) Discriminant (A): A = b? - 4ac + A>0O: Two distinct real roots. A = 0: Two equal (repeated) real roots. A <0: Two complex (imaginary) conjugate roots. IV. Relationship Between Roots and Coefficients Let a and B be the roots of ax? + bx +c =0. Sum of Roots (a + B): a + B = -b/a Product of Roots (af): aB = c/a V. Forming a Quadratic Equation Given the Roots If a and B are the roots, the quadratic equation is: x?-(a+B)x+aB =0 + x?-(Sum of Roots)x + (Product of Roots) = 0 * Trick: This is super useful for quickly building equations when you know the roots. VI. Short Tricks and Problem-Solving Tips + Symmetric Roots: Equation form : ax? + c =0 Reciprocal Roots: If one root is the reciprocal of the other, then c = a. Common Root: If two quadratic equations have a common root, solve the equations simultaneously to find the common root and the relation between their coefficients. Sign Analysis: Use the signs of the coefficients to infer information about the location and nature of the roots. (e.g., If ‘a’ and 'c' have opposite signs, the roots are real and distinct.) « Transformations of Roots: 
