DOC-20230724-WA0004.

[Audio] Quadratic equations. JeodDZ. Quadratic equations.

[Audio]  ❖Quadratic equation (a + b) = a2 + b2 + 2(a)(b) (a+b)3 = a3 + 3a2b + 3ab3 + b3 ❖ roots of quadratic equation ❖ nature of quadratic equation ❖ discriminant formula ❖ solution of the roots ❖ mcq's , questions & case study INTRODUCTION.

[Audio]   The standard form of quadratic equation is :  ax2 + bx + c = 0  or we can say whose highest power\degree is 2  Where a, b, c are real numbers and a ≠ 0 QUADRATIC EQUATION.

[Audio]  FOR EXAMPLE (x + 1)2 = 2(x – 3) ➢ ( x )2 + ( 1 )2 + 2(x)(1) = 2x – 6 ➢ x2 + 1 + 2x = 2x – 6 ➢ x2 + 1 + 2x – 2x + 6 = 0 ➢ x2 + 7 = 0 (a + b) = a2 + b2 + 2(a)(b) This is a quadratic equation as it has 2 as its highest degree.

[Audio]  (x+ 2 ) 3 = 2x(x2-1) ➢ (x)3 + 3(x)2(2) + 3(x)(2)2 + (2)3 = 2x3 – 2x ➢ x3 + 6x2 + 12x + 8 = 2x3 – 2x ➢ 2x3 – 2x - x3 - 6x2 - 12x - 8 = 0 ➢ X3 -6x2 – 14x – 8 = 0 It is in cubic form not in a quadratic form (a+b)3 = a3 + 3a2b + 3ab3 + b3.

[Audio]   middle term splitting = factorisation zeroes of the quadratic equation and roots of quadratic equation is same Let's see some examples Root of the quadratic equation.

[Audio]  100x2 – 20x + 1 = 0 ➢ 100x2 - 10x – 10 x + 1 = 0 ➢ 10x(10x – 1 ) – 1 ( 10x – 1) ➢ (10x -1) (10x -1) 10x -1= 0 10x -1= 0 X= 1\ 10 X= 1\ 10 X= 1\ 10 , X= 1\ 10 are the roots of the x ❑ Multiply first and last term = 100 ❑ 10 × 10 = 100 ❑ 10 + 10 = 100.

[Audio]  Discriminant formula Nature of roots D = b2 – 4ac If D = 36 If D = -36 If D = 36 - 36 Then D > 0 Then D < 0 Then D = 0 D = +ve D = -ve NATURE = Real and distinct NATURE = no real root \ root does not exist NATURE = two equal roots.

[Audio]  Formula D = B2 – 4ac ❖A = 2 , B = 2 , C = 3 ❖(2)2 – 4(2)(3) = 0 ❖4 – 24 ❖-20 ❖D <0 It has no real root 2x2 + 2x + 3 = 0.

[Audio]  Solution of the root X2- 3x – 4 = 0 First we solve D (b2 – 4ac) o (-3)2 – 4(1)(-4) = 0 o 9 + 16 = 0 o 25 o -(-3) ± √25 2(1) o 3 ± 5 2.

[Audio]  1. Which of the following is not a quadratic equation A) 2(x-2)2 = 4x2 – 2x + 1 B) 2x- x2 = x2 – 5 C) (√2x + √3)2 +x2 = 3x2 – 5x D) (x2 + 1) = x4 + 3 + 4x2 2 . If ½ is the root of the equation x2 + kx 5 4 = 0 A) ½ B) -2 C) 1 4 D) 2 3. A quadratic equation can have : A) At least two roots B) at most two roots C) Always two roots D) only one root MCQ′S.

[Audio]  4) IF D > 0 then what will be the nature of the root A) No real root B) real and distinct C) Two equal roots D) both B) and C) 5) The quadratic equation 3x2 – 5x + 2 = 0 has : A) D > 0 B) D < 0 C) D = 0 D) D ≠ 0 6) IF the zeroes of the quadratic polynomial p(x) = ax2 + bx + c = 0, a ≠ 0 are 4 , -5 then the roots of the quadratic equation are : A) 4, -5 B) -4,5 C) D) , MCQ′S.

[Audio]  7) If the product of two consecutive odd positive integers is 323 , then the integers are A) 21,13 B) 19,17 C) 31,33 D) 29,27 8) IF the discriminant of the quadratic equation = ax2 + bx + c = 0 is zero then the roots of the equation : A) are rational and equal B) are irrational and equal C) Are real and equal D) do not exit in real MCQ′S.

[Audio]  1.C) 5. A 2.D) 6. A 3.B) 7. B 4. B 8. ANSWERS.

[Audio]  1. DETERMINE THE VALUE OF K IN THE QUADRATIC EQUATION 4X2 – 3KX + 1 = 0 HAS EQUAL ROOTS 2. SOLVE USING DISCRIMINANT METHOD 9X2 – 12X + 4 = 0 3. DETERMINE THE VALUE OF 'P' PX2 + 4X + 1 = 0 4. FIND THE SOLUTIONS OF QUADRATIC EQUATION 2X2 –X-6 = 0 5. TELL WHETER X3 – 4X2 – X + 1 = ( X – 2)3 IS QUADRATIC OR NOT QUESTIONS.

[Audio]  1.K = ± 4/3 2.X = 2/3 3.P = 4 4.X = 2 5.YESS ANSWERS.

[Audio]  RAJ AND AJAY GO TO HIMACHAL BY THEIR OWN CARS. RAJ'S CAR TRAVELS AT A SPEED OF X WHILE AJAY'S CAR TRAVELS 5 KM/H FASTER THAN RAJ'S CAR. RAJ TOOK 4 HOURS MORE THAN AJAY TO COMPLETE THE JOURNEY OF 400 KM. 1) WHAT WILL BE THE DISTANCE COVERED BY AJAY'S CAR IN TWO HOURS ? A) 2(X + 5)KM B) (X-5)KM C) 2(X+10)KM D) ( 2X + 5)KM CASE STUDY.

[Audio]  2)Which of the following quadratic equation describe the speed of raj's car? A) X2 – 5X –5OO = 0 B) X2 + 4X – 400 = 0 C) X2 + 5X – 500 = 0 D) X2 – 4X + 400 = 0 3) WHAT IS THE SPEED OF RAJ 'S CAR ? A) 20 B) 15 C) 25 D) 10 4) HOW MUCH TIME TOOK AJAY TO TRAVEL 4OO KM ? A) 40H B) 20H C) 25 H D) 16H QUESTIONS.

[Audio]  1.A 2(X+5)KM 2.C – X2 + 5X – 500 = 0 3.A - 20 4.D - 16 ANSWERS.