QUADRATIC EQUATION-TIPS

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IMPORTANT TIPS

• ax² + bx + c is a quadratic polynomial or second degree polynomial where a, b, c are real constants and a ≠ 0.



• x² - √ 5 x + 7, 3x² - 2x + 1, x² - 3, 4x², etc. are examples of second degree polynomials.



• The values of the variable(x) for which the value of the polynomial is zero, are called ZEROS OF A POLYNOMIAL.



• For a second degree polynomial we can have at the most two values of the variable for which the value of the polynomial is zero, i.e. there are at the most two zeros of a second degree polynomial.



• For p(x) = x² - 6x + 8, we have x = 4 and x = 2 for which the value of p(x) is 0. Thus, 4 and 2 are the zeros of p(x).



• For p(y) = y² + 4, we cannot find a value of y for which p(y) is 0. Thus, a polynomial may not have zeros.



• If p(x) is a second degree polynomial, then p(x) = 0 is called a QUADRATIC EQUATION.



• ax² + bx + c = 0 is the general form of a quadratic equation, where a, b, c ∈ R and a ≠ 0.



• The values of x which satisfy the equation ax² + bx + c = 0 (a ≠ 0) are called the roots of that quadratic equation.



• Zeros of polynomial ax² + bx + c are the roots of the quadratic equation ax² + bx + c = 0.



• If ax² + bx + c = (x - α) (x - β) then α and β are the roots of the equation ax² + bx + c = 0.



• All polynomials of the form ax² + bx + c cannot be factorized by normal methods of factorization into factors like (x - α) and (x - β). In such cases, equation ax² + bx + c = 0 is solved by the METHOD OF PERFECT SQUARE ( also known as the Method of Discriminant or the Method of formula).



• Method of Perfect Square can be used to solve any quadratic equation which has a real solution, i.e. solution in R.



• By the method of perfect square, we obtain the roots α and β of the quadratic equation ax² + bx + c = 0 (a ≠ 0) as
  
  α = ( - b - √ D)/2a
  
  β = ( - b + √ D)/2a
  
  where D = b² - 4ac and it is called the DISCRIMINANT of the given quadratic equation.



• If D ≥ 0, the roots of the equation are real.



• If D > 0, the roots are real and distinct.



• If D > 0, D is not a perfect square ( for example : D = 22), then the roots are irrational and distinct.



• If D > 0, D is a perfect square ( for example : D = 25), then the roots are rational and distinct.



• If D = 0, the roots 0f the equation are equal ( also called 'repeated' or 'identical' roots), i.e. the equation has only one root. The value of this root is ( - b )/2a .



• If D < 0, ( i.e. D is negative ), then the given equation has no real roots. ( We say real roots do not exist ).