Central Public School,Samastipur And Sadhana Devi Vidyapith, Samastipur [Student Practice]
Quadratic Equations : Contents
1. Standard form of a quad. Equations: ax2 + bx + c = 0 (a ≠ 0)2. Solution of the quadratic equations only real roots by
(a) Factorization (b) Completing the square (c) Quadratic formula
3. Relationship between discriminant and nature of roots.
if D > 0 => Real & Distinct Roots if D = 0 => Real & Equal Roots if D < 0 => No Real Roots
4. Problems related to day to day activities
Learning Objective: To learn the following facts
1. A second degree Equation of the form ax2 + bx + c = 0 where a, b, c are real numbers and a ≠ 0 are called quadratic eqn.
2. If ax2 + bx + c = 0 equivalent to (x – a) (x – b) = 0 then x = a and x = b is a solution.
3. If ax2 + bx + c = 0 has a & b as roots of eqn then sum of roots = – b/a. Product of roots = c/a.
4. (b2 – 4ac) is called the discriminant of the eqn and
1. Roots of an eqn. 2. Discriminant 3. Real & Non Real Roots
4. Quadratic Formula
Practice PaperQ. Which constant should be added and subtracted to solve the quadratic equation 4x2 - Ö3x -5=0 by the method of completing the squares ?
(A) 9/16 (B) 3/16 (C)3/4 (D) Ö3/4
Q. If the discriminant of the equation 6x2-bx+2=0 is 1, then the value of ‘b’ is :
(A) 7 (B) -7 (C) ±7 (D) ±Ö7
Q. The value of k for which roots of the quadratic equation kx2+ 2x+3= 0 are equal is :
A) 1/3 (B) - 1/3 (C) 3 (D) -3
Q. ax2+ bx+ c= 0, a > 0, b= 0, c > 0 has
(A) two equal real roots (B) one real root (C) two distinct real roots (D) no real rootsQ. If the roots of the equation 12x2+ mx+ 5= 0 are real and different then m is equal to :
(A) 8Ö15 (B) 2Ö15 (C) 4Ö15 (D) 10Ö5
Q. Which of the following equations has the sum of roots as 3 ?
(A) 2x2- 3x+6 =0 (B) -x2 + 3x -3 = 0 (C) Öx2 – (3/Ö2)x + 1 = 0 (D) 3x2 - 3x -3 = 0
Q . Which of the following is a quadratic equation?
(i) x2+5x +6 = 0 (ii) 3x+ 4/Öx + 5 = 0 (iii) Öx + 2x + 3= 0 (iv) 3x2 – 27/x = 0
Q. The value of(a) –2 (b) –4 (c) 2 (d) 4 Q. If the equation kx2-2kx+6=0 has equal roots, then find the value of k.
Q. Find the roots of the following quadratic equation x2- ( x/5 ) + (1/100) = 0
Q. Find the roots of the equation ax2+ a=a2x+x
Q. Solve for x : 4Ö3 x2+ 5x-2 =0
Q. Solve for x : 2Q. Solve for x : ax2+ (4a2- 3b)x-12ab = 0
Q. Solve the quadratic equation 9x2 - 15x+ 6= 0 by the method of completing the square
Q. Solve for x :Ö5 x2- 3 Ö6 x- Ö20 =0
Q. Solve for x : 9x2 - 9(a+ b)x+ [2a2+5ab+2b2] = 0.
Q. Solve for x, 12abx2 - (9a2 +8b2)x + 6ab = 0.
Q. Find the roots of the quadratic equation :
(1/3)x 2 - Ö11 x + 1 = 0
Q. Solve for x :
Q. Solve for x: [Ans: x=-a ; or, x =- b/2 ]