Page 1 :
CLASS X : CHAPTER - 4, QUADRATIC EQUATIONS, , IMPORTANT FORMULAS & CONCEPT:, , POLYNOMIALS, , An algebraic expression of the form p(x) = ap + ajx + ox? + 3X9 4 Lecce apx", where a # 0, is, called a polynomial in variable x of degree n., , ete sap; 81,82, Bay seceseces ,@y are real numbers and each power of x is a non-negative integer., , e.g. 3x” — 5x + 2 is a polynomial of degree 2., 3Vx +2 is nota polynomial., , > If p(x) is a polynomial in x, the highest power of x in p(x) is called the degree of the polynomial, p(x). For example, 4x + 2 is a polynomial in the variable x of degree 1, ay -3y +4 isa, polynomial in the variable y of degree 2,, , * A polynomial of degree 0 is called a constant polynomial., , “A polynomial p(x) = ax + b of degree | is called a linear polynomial., , “A polynomial p(x) = ax” + bx + c of degree 2 is called a quadratic polynomial., “A polynomial p(x) = ax? + bx? + cx + dof degree 3 is called a cubic polynomial., , “A polynomial p(x) = ax* + bx* + cx? + dx +e of degree 4 is called a bi-quadratic polynomial., , QUADRATIC EQUATION, , A polynomial p(x) = ax’ + bx +c of degree 2 is called a quadratic polynomial, then p(x) = 0 is, known as quadratic equation., , e.g. 2x” — 3x + 2 = 0, x? + 5x + 6 = 0 are quadratic equations., , METHODS TO FIND THE SOLUTION OF QUADRATIC EQUATIONS, Three methods to find the solution of quadratic equation:, , 1. Factorisation method, , 2. Method of completing the square, , 3. Quadratic formula method, , FACTORISATION METHOD, , Steps to find the solution of given quadratic equation by factorisation, , > Firstly, write the given quadratic equation in standard form ax” + bx + c = 0., , > Find two numbers @ and # such that sum of @ and # is equal to b and product of @ and # is, equal to ac., , > Write the middle term bx as ax+ x and factorise it by splitting the middle term and let factors, are (x + p) and (x + q) i.e. ax’ +bx+c0=0> (x + p(x +q)=0, , > Now equate reach factor to zero and find the values of x., , > These values of x are the required roots/solutions of the given quadratic equation., , METHOD OF COMPLETING THE SQUARE, Steps to find the solution of given quadratic equation by Method of completing the square:, > Firstly, write the given quadratic equation in standard form ax” + bx + c = 0., > Make coefficient of x” unity by dividing all by a then we get, *, 0 c, x +—X+—=, aoa

Page 2 :
> Shift the constant on RHS and add square of half of the coefficient of x i.e. (2) on both sides., , MA, »b ae b bY _c (by, x +Sx=S——S 2" +2| — [xt =-—+| —, a a 2a 2a a \2a, , > Write LHS as the perfect square of a binomial expression and simplify RHS., , 2, , ( b J b? —4ac, x+—] =——, 2a 4a”, , > Take square root on both sides, , ea aa lb 4ac, 2a, , > Find the value of x by shifting the constant term on RHS ie. x= + p= Sac._b, , 4a 2a, , QUADRATIC FORMULA METHOD, , Steps to find the solution of given quadratic equation by quadratic formula method:, , > Firstly, write the given quadratic equation in standard form ax” + bx + c = 0., , >» Write the values of a, b and c by comparing the given equation with standard form., , > Find discriminant D = b’ — 4ac. If value of D is negative, then is no real solution i.e. solution, does not exist. If value of D2 0, then solution exists follow the next step., , ‘ ‘ -b+ JD ‘, > Put the value of a, b and D in quadratic formula oe and get the required, a, roots/solutions., NATURE OF ROOTS, , The roots of the quadratic equation ax’ +bx+c¢=0 by quadratic formula are given by, , _-b+Vb?-4ac _ -b+JD, - 2a ~ 2a, where D = b?—4ac is called discriminant. The nature of roots depends upon the value of, discriminant D. There are three cases —, Case -I, When D > 0 i.e. b> —4ac > 0, then the quadratic equation has two distinct roots., , -b+VD -b-VD, , ie. x =————_ and, , 2a 2a, Case - II, When D = 0, then the quadratic equation has two equal real roots., —b —b, ie. x=— and —, 2a 2a, Case - III, , When D < 0 then there is no real roots exist.

Page 3 :
Previous Year Boards Questions, Chapter 4 - Quadratic Equations, , 1 Mark:, 1. If one root of the quadratic equation 6x* — x — k = Ois 5 then find the value of k. CBSE 2017, Foreign (30/2/1), 2. Find the value of k, for which one root of the quadratic equation kx? — 14x + 8 = 0 is six times the other., , 10., , 11., , 12., 13,, , 14., , CBSE Sample Paper 2016, , Ifx = -5 is a solution of the quadratic equation 3x? + 2kx — 3 = 0, find the value ofk. | CBSE 2015, Delhi (30/1/1), , If the quadratic equation px? — 2V5 px + 15 = 0 has two equal roots, then find the value of p., , CBSE 2015, Outside Delhi (30/1), If 1 is a root of the equations ay” + ay + 3 = 0 and y* + y + b = 0, then ab equals:, 7, A) 3 By) =3, C) 6 D) -3, CBSE 2012, Delhi (30/1/1), If the quadratic equation mx* + 2x + m = 0 has two equal roots, then the values of m are, A) +1 B) 0,2 C) 0,1 D) -1,0, CBSE 2012, Foreign (30/2/1), The roots of the quadratic equation 2x” — x — 6 = O are, A) -2,3/2 B) 2,-3/2 ©) -2,-3/2 D) 2,3/2, CBSE 2012, Outside Delhi (30/1), The roots of the equation x? + x — p(p + 1) = 0, where p is a constant, are, A) pp+1 B) -p,p+1 ©) p-@+1) D) -p,-(p +1), CBSE 2011, Delhi (30/1/1), The roots of the quadratic equation x* + 5x — (@ + 1)(a + 6) = 0, where @isa constant, are, A) a+1,a+6 B) (a+1),-—(a +6), C) -(a+1),(@ +6) D) —(a@ +1),—(a + 6), CBSE 2011, Foreign (30/2/1), The roots of the equation x* — 3x — m(m + 3) = 0, where mis a constant, are, A) mm+3 B) -—m,m+3 C) m,—(m+3) D) —m,—(m + 3), CBSE 2011, Outside Delhi (30/1), Find the discriminant of the quadratic equation, 3V3x? + 10x + V3 = 0. CBSE 2009, Outside Delhi (30/1), Write the nature of roots of quadratic equation 4x? + 4V3x + 3 = 0. CBSE 2009, Foreign (30/2/1), For what value of k the quadratic equation x? — kx + 4 = 0 has equal roots? CBSE Sample Paper I 2008, , What is the nature of roots of the quadratic equation 4x* — 12x — 9 = 0? CBSE Sample Paper II 2008

Page 4 :
2 Marks:, , ae, , 10., , 11., , 12., , 13., , 14., , 15., , 16., , 17., , 18., , 19., , 20., , 21., , 22., , Find the value of p, for which one root of the quadratic equation px? — 14x + 8 = 0 is 6 times the other., , CBSE 2017, Outside Delhi (30/1), Find the roots of the quadratic equation V2x? + 7x + 5V2 = 0. CBSE 2017, Delhi (30/1/1), Find the value of k for which the equation x? + k(2x + k — 1) + 2 = Ohas real and equal roots., CBSE 2017, Delhi (30/1/1), Solve for x:, V3x? + 10x — 8v3 = 0. CBSE 2017, Foreign (30/2/1), , If —5 is a root of the quadratic equation 2x? + px — 15 = 0 and the quadratic equation p(x? + x) + k = 0 has equal, roots, find the value of k. CBSE 2016, Outside Delhi (30/1), , Ifx= Zand x = —3 are roots of the quadratic equation ax? + 7x + b = 0, find the values of a and b., , CBSE 2016, Delhi (30/1/1), A two digit number is four times the sum of the digits. It is also equal to 3 times the product of digits. Find the number., CBSE 2016, Foreign (30/2/1), If 2 is a root of the equation x? + kx + 12 = 0 and the equation x* + kx + q = 0 has equal roots, find the value of q., CBSE Sample Paper 2016, Solve the following quadratic equation for x:, 4x? — 4a?x + (a* — b*) =0 CBSE 2015, Delhi (30/1/1), Solve for x:, x? —(v3+1)x+V3=0 CBSE 2015, Foreign (30/2/1), Solve the following quadratic equation for x:, 4x? + 4bx — (a? — b*) =0 CBSE 2015, Outside (30/1), Find the roots of the quadratic equation 3x* — 2V6x + 2 = 0. CBSE Sample Paper 2015, , Find the values of p for which the quadratic equation 4x? + px + 3 = 0 has equal roots. CBSE 2014, (30/1), (30/3), Find the values of k for which the quadratic equation 9x? — 3kx + k = 0 has equal roots. CBSE 2014 (30/2), (30/3), , Solve the following quadratic equation for x:, 4v3x? + 5x —2V3 =0 CBSE 2013, Delhi (30/1/1), , Find the value(s) of k so that the quadratic equation x* — 4kx + k = 0 has equal roots. | CBSE 2012, Delhi (30/1/1), , Find the value of k for which the roots of the quadratic equation (k — 4)x? + 2(k — 4)x + 2 = 0 are equal., CBSE 2012, Foreign (30/2/1), , Find the value of p.for which the roots of the equation px(x — 2) + 6 = 0, are equal. CBSE 2012, Outside Delhi (30/1), Find the value of p so that the quadratic equation px(x — 3) + 9 = 0 has two equal roots. CBSE 2011, Delhi (30/1/1), , For what value of k does the quadratic equation (k — 5)x? + 2(k — 5)x + 2 = 0 have equal roots?, , CBSE 2011, Foreign (30/2/1), Find the roots of the following quadratic equation:, V3x? — 2V2x - 2V3 =0 CBSE 2011, Foreign (30/2/1), , Find the value of m so that the quadratic equation mx(x — 7) + 49 = 0 has two equal roots., CBSE 2011, Outside Delhi (30/1)

Page 5 :
3 Marks:, , 1, , 10., , i., , 12., , 13., , 14., , 15., , 16., , a7., , 18., , 19., , 20., , 21., , 22., , If ad # bc, then prove that the equation, , , , (a? + b?)x? + 2(ac + bd)x + (c? + d*) = 0 has no real roots. CBSE 2017, Outside Delhi (30/1), If the equation (1 + m?)x? + 2mcx + c? — a? = 0 has equal roots then show that c? = a?(1 + m?)., CBSE 2017, Delhi (30/1/1), , If the roots of the quadratic equation (a — b)x? + (b —c)x + (c — a) = O are equal, prove that, , CBSE 2017, Foreign (30/2/1), , =—4 —, , Solve for x = oy + Ginasn Tart #123 CBSE 2016, Outside Delhi (30/1), Solve for x:, f+s iO = 0,x # 3,-3/2 CBSE 2016, Delhi (30/1/1), , X-3 © 2x+3 — (x—3)(2x+3), , Solve the given quadratic equation for x : 9x* — 9(a + b)x + (2a +5ab+2b*)=0. CBSE 2016, Foreign (30/2/1), , i. —@ 878, Solve iam Tate t zat bh #0. CBSE Sample Paper 2016, Find that non-zero value of k, for which the quadratic equation kx? +1-2(k-—1)x +x? = 0 has equal roots. Hence, find the roots of the equation. CBSE 2015, Delhi (30/1/1), Solve for x:, , x? +5x—(a?+a—-6)=0 CBSE 2015, Foreign (30/2/1), Solve for x:, , V3x? — 2V2x - 2v3 = 0 CBSE 2015, Outside Delhi (30/1), , Solve the following quadratic equation for x, x? — 4ax — b? + 4a? =0 CBSE 2012, Outside Delhi (30/1), , If the sum of two natural numbers is 8 and their product is 15, find the numbers. CBSE 2012, Outside Delhi (30/1), , If a student had walked 1 km/hr faster, he would have taken 15 minutes less to walk 3 km. Find the rate at which he, , was walking. CBSE Sample Paper III 2008, Find the value of k so that the following quadratic equation has equal roots:, , 2x? —(k-—2)x+1=0 CBSE Sample Paper III 2008, Solve the following equation for z., , s-waiee 1,0,—2 CBSE 2015, Sample Paper 2015, Solve for x:, , 4-1-5; x#0,-1 CBSE 2014 (30/1), (30/2), (30/3), , For what value of k, are the roots of the quadratic equation kx(x — 2) + 6 = 0 equal? CBSE 2013, Delhi (30/1/1), , Solve for x : 4x? — 4ax + (a? — b?) =0 CBSE 2012, Delhi (30/1/1), Solve for x : 3x? — 2V6x +2 =0 CBSE 2012, Delhi (30/1/1), Solve for x:, , 4V3x? + 5x — 2V3 =0 CBSE 2012, Foreign (30/2/1), , Solve the following quadratic equation for x, x* — 4ax — b* + 4a7 =0 CBSE 2012, Outside Delhi (30/1), , If the sum of two natural numbers is 8 and their product is 15, find the numbers. CBSE 2012, Outside Delhi (30/1)