QuadraticEquations slides 617 kho tài liệu bách khoa

Quadratic Equation MC Sir Introduction, Graphs Inequality Theory of Equations : Relation between Roots and Coefficients of Cubic and Higher Polynomials Identity Infinite Roots, Common Roots Maximum and Minimum Values of Quadratic and Rational Function Quadratic Equation MC Sir General 20 in x and y Condition for General 20 in x and y to be factorized in two linears Location of Roots 10.Modulus Inequality 11.Logarithm Inequality MC Sir Quadratic Equation No of Questions 2008 2009 2010 2011 2012 Quadratic y = ax2 + bx + c ; a a = leading coefficient b = coefficient of linear term c = absolute term y = f(x) = ax2 + bx + c In case a=0 y = bx + c is linear polynomial a=c=0 y = bx is odd linear polynomial Cubic Polynomial y = ax3 + bx2 + cx + d a = leading coefficient d = absolute term Roots of Quadratic Equation y = ax2 + bx + c Where D = b2 – 4ac is called discriminant ax2 + bx + c = Sum of roots = – b/a Product of roots = c/a D = b2 – ac Different Graphs of Quadratic Expression Example Q y = x2 + 2x + = (x + 1) + y Parabola D=2 –8=–4 10 17 ∞ ∞ 10 Example Q Show that for any real value of a (a2 + 3) x2 + (a + 2) x – < is true for at least one negative x 239 Example Q If f(x) = 4x + ax + (a – 3) is negative for at least one negative x, find all values of a 240 Example Q Find a for which x + 2(a – 1) x + a + = has at least one positive root 241 Example Q Find p for which the least value of 4x2 – 4px + b2 – 2p + in x [0,2] is equal to 242 Example Q Find k for which the equation x4 + x2 (1 – 2k) + k2 – = has (i) No real solution 243 Example Q Find k for which the equation x4 + x2 (1 – 2k) + k2 – = has (ii) one real solution 244 Example Q Find k for which the equation x4 + x2 (1 – 2k) + k2 – = has (iii) two real solution 245 Example Q Find k for which the equation x4 + x2 (1 – 2k) + k2 – = has (iv) three real solution 246 Example Q Find k for which the equation x4 + x2 (1 – 2k) + k2 – = has (v) Four real solution 247 Modulas Inequality 248 Example Q 249 Note |x|< |x|> x (- , ) x (- , - ) ( , ) 250 Example Q (| x – | – 3) (| x + | – 5) < 251 Example Q | x – 5| > | x2 – 5x + | 252 Example Q 253