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|, |, , Deines An equation of the form ax’ + bx +¢=0, fi ax’ + bx +c =0, whereaz+0, 1. Speed= —Fae where a,b,c are real numbers and a + 0, is D = (b’ - 4ac). and the roots are, called a quadratic equation in x. i given by, , 2. Area of figures, , , , 3. Flow rate x time = volume of water Solution or Roots of Quadratic Equation pe eo vb and B =zb=v0, 4. Number or ages A real number « is called a root of the 2a 2a, quadratic equation ax’ + bx +c = 0,a#O if Case - I, ao? +ba+c=0 a When D > 0, roots are real distinct, and given by, e.g. The hypotenuse of a right triangle is 25 cm. acee ND and pa vo, The difference between the lengths of the other two Method at Ws gases 2 2a, , In this method (ax’ + bx + c) be expressible, as the product of two linear expression , say, (px + q) and (rx + s), where p,q, r are real, numbers such that p #O andr #0, , Then ax’ + bx +c = 0 = (px + q) (rx +s)=0, , , , , , , , , , , Application Nature of roots, , , , , , , , , , , , , , , , , , , , , , , , , , , , , sides of the triangle is 5 cm. Find the lengths of these sides., , When D = 0, roots are real and equal, Sol. Shorter side = x cm, longer side = (x + 5) cm., , finding solution ahd toate ee aueh By, , hypotenuse = 25 cm a=B=- 2., by pythagoras Theorem = | 2a, x’ + (x +5) = 25° Completing the square method meters, , x’ + 5x - 300=0 _ When D < 0, roots are not real., , (x + 20)(x - 15)=0, , This gives x = 15 or x = - 20., , We reject x = - 20 and take x = 15., , Thus, length of shorter side = 15 cm., , Length of longer side = (15 + 5) cm, i.e., 20 cm., , ax’ +bx+c=0,a0., , bos, x7 4+=x+—=0, aa, , Factorisation method, , Quadratic formula :, for ax’ + bx+c=0,, , , , , , = (px + q)= 0 or (rx +s)=0 D=b’ - 4ac i, s, axs-3 Ree Obtain the values of x by shifting =r bevD, the constant term oe on RHS 2a }, , NCERT / X / Quadratic Equation