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[email protected] TARIYAL CLASSES CONTACTS NO: 9871845767, , MATHEMATICS (041), QUADRATIC EQUATION, , Question 1: Find two consecutive numbers whose squares have the sum 85., , Question 2: Divide 29 into two parts so that the sum of the squares of the parts is 425., Question 3: Two squares have sides x cm and (x + 4) cm. The sum of their areas is 656 cm’., Find the sides of the squares., , Question 4: The sum of two numbers is 48 and their product is 432. Find the numbers., Question 5: If an integer is added to its square, the sum is 90. Find the integer with the help of, quadratic equation., , Question 6: Find the whole number which when decreased by 20 is equal to 69 times the, reciprocal of the number., , Question 7: Find two consecutive natural numbers whose product is 20., , Question 8: The sum of squares of two consecutive odd positive integers is 394. Find them., Question 9: The sum of two numbers is 8 and 15 times the sum of their reciprocals is also 8., Find the numbers., , Question 10: The sum of a number and its positive square root is 6/25. Find the number., Question 11: The sum of a number and its square is 63/4, find the numbers., , Question 12: There are three consecutive integers such that the square of the first increased, by the product of the other two gives 154. What are the integers?, , Question 13: The product of two successive integral multiples of 5 is 300. Determine the, multiples., , Question 14:The sum of the squares of two numbers is 233 and one of the number is 3 less, than twice the other number. Find the numbers., , Question 15: Find the consecutive even integers whose squares have the sum 340., Question 16:The difference of two numbers is 4. If the difference of their reciprocal is 4/21,, find the numbers., , Question 17: Find two natural numbers which differ by 3 and whose squares have the sum 17., Question 18: The sum of the squares of three consecutive natural number is 149. Find the, numbers., , Question 19: The sum of two numbers is 16. The sum of their reciprocals is 1/3. Find the, numbers., , Question 20: Determine two consecutive multiples of 3 whose product is 270., , , , , , HINT.1. Let two consecutive numbers i a Let first numbers be x and, be xand (**!) other (79~*), Then according to question Then according to question., x? +(x+1) =85 x°+(29-x) =425, , x7 4x7 4+274+1=85 x? +x° —58x +841 = 425, , 2x? +2x-854+1=0 2x? —58x+841=425, 2x? +2x-84=0 2x° —58x+841—425=0, 2x? —58x+416=, , , , , , , , Sate [email protected] CONTACT NO:9871845767

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[email protected], , TARIYAL CLASSES CONTACTS NO: 9871845767, , , , HINT.3 Given that the sides of two square, , be xem and (**4)em, Then according to question, , x? 4+(x+ 4y = 656, , x7 4+ x7 +4+8x+16=656, 2x° +8x+16—656=0, 2x? +8x—640=0, , HINT.4 Let first numbers be x and, , other (48-*), , Then according to question, x(48—x) = 432, 48x—x? = 432, , x’ —48x+432=0, , , , | HINT.5 Let an integer be x., Then according to question, , x+x° =90, x +x-90=0, x? +10x-9x-90=0, , HINT.6 Let the whole numbers be x., Then according to question, , (x-20)- 69x4, x, x(x-20)=69, x? -20x-69=0, , , , , , , , HINT.7 Let two consecutive numbers, , , , HINT.8 Let two consecutive odd positive, , , , (x+1) F (2x-1) (2x +1), be x and integer be and other, Then according to question Then according to question, , x(x+1)=20 (2x+1) +(2x-1) =394, x'+x-20=0 4x + A¥ +14+4x° 24¥ +1=394, x +5x-4x-20=0 ;, 8x? +2 =394, x(x+5)—4(x+5)=0, (x+5)(x-4)=0, HINT.9 : Let first numbers be x and HINT 10 :Let first numbers be x, (8—x) Then according to question, , other, Then according to question, , , , , , 36 (8—x) |, aeeeh, | x(8-) 15, f i 1d, [x(8-x)] 15 ¥, , , , “duleeni™, , Xx+VX= 25, , Let *=" then, 2 6, , y+y=—, , ~ ~ 28, , , , , , ae, , [email protected], , CONTACT NO:9871845767

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[email protected], , TARIYAL CLASSES CONTACTS NO: 9871845767, , , , HINT.11 :Let first numbers be x, Then according to question, , , , > 63, X+xX =, 4, Let *=)" then, 4(x+x°)=63, 4x*+4x-63=0, , 4x’ +18x-14x-63=0, , HINT.12: Let three consecutive integer, , be (2+) ang (+2), Then according to question, x? +(x41)(x+2)=154, , x 4+x743x42=154, 2x? +3x+2-154=0, , 2x? +3x-152=0, , , , HINT.13 :Let the successive integer multiples, Sx-4 :, of 5 be 5x, and 5(x +1), Then according to question, 5xx5(x+1)=300, 25{ x* +x) =300, x +x=12, , x +x-12=0, , HINT.14: Let the numbers be integers. One of, the numbers be x. So, the other will be (7*~>), Then according to question,, x? +(2x-3) =233, x? 44x? -12x +9 =233, 5x° -12x+9-233=0, , 5x? -12x-224=0, , , , HINT.15 :Let two consecutive even integer, be 2* and other(2**2), Then according to question, (2x) +(2x+2) =340, 4x? +4x° +8x4+4=340, 8x* +8x =340-4, 8x" +8x—336=0, , HINT 16 : Let one numbers be x then other, (x+4), , Then according to question, , 1 1 4, x (x+4) 21, A+4-f£ 4, x(x+4) 21, , 4 4, (x? + 4x) “21, , , , , , HINT 17: Let one natural number be * and, other(*~>),, Then according to question, (x) +(x-3) =117, x? +x? -6x49=117, 6x+9-117=0, , 2x" -6x-108=0, , , , , , HINT 18: Let three consecutive integer, be x, (x+1) and (**2), Then according to question, x° +(x+1) +(x+2) =149, x + 2x41 4x7 4+4x4+4=149, 149=0, 144=0, , , , 3x7 + 6x45, , 3x° + 6x, , , , , , ae, , [email protected], , CONTACT NO:9871845767

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[email protected] TARIYAL CLASSES CONTACTS NO: 9871845767, , , , , , HINT 19 :Let one numbers be x then other HINT 20 :Let the required number, (16 x). be 3x and (3x+3), Then according to question Then according to question, ee (3x)(3x+3)=270, x (16—x) 3 9x? +9x-270=0, 16-£+4 _| 9(x? +x-30)=0, x(16-x) 3 5, x°+x-30=0, 16 1, (16x—x") 3, , , , , , , , TARIYAL CLASSES 2020-2021 (ONE YEAR AGO), , , , 100% IMPROVEMENT RESULT CONTACT NO.9871845767, , Sate [email protected] CONTACT NO:9871845767

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Quadratic Equations, , , , Topics Covered, , 1. Quadratic Equations, , 2. Solution of a Quadratic Equation by Factorisation, , 3. Solution of a Quadratic Equation by Quadratic Formula, 4. Nature of Roots, , , , , , , , 1. Quapratic Equations, , The second degree polynomial equations is commonly known as quadratic equation, 1.e., if p(x) is a quadratic polynomial,, then p(x) = 0 is called a quadratic equation. The general form of quadratic equation in the variable x is ac+bx+c=0,, where a, b, c are real numbers and a = 0., , For example, 2x? +x- 150 =0, 3x” —2x+5=0, 4x—-3x° + 2=O0are quadratic equations., Example 1. Check whether the following are quadratic equations:, , , , , , (a) x(x +2)-3=(x +4) x (b) @ +2 =x - 4742, Solution. (a) Since x(x +2)—3 =x(x+4), > VP +2x—3 =x? + dx, =. 2x+3=0, , This is linear equation not a quadratic equation, (b) (xt 2p =x 4°42, > P+ 6x? + 12+ 8 = 47 +2, > 10x? + 12x + 6 =, = Sx + 6x +3 =0, This is a quadratic equation., Example 2. Represent the following situation in the form a quadratic equation:, (a) Abdul and Sneha together have 30 oranges. Both of them ate 3 oranges each and the product of the number of, oranges they have now is 120. We would like to find out how many oranges they had initially., (6) The area of a rectangular plot is 428 m?. The length of the plot (in metres) is two more than twice its breadth. We, need to find the length and breadth of the plot., Solution. (a) Let the number of number of left oranges with Abdul and Sneha be x and y respectively, Then, x+y=30 > y=30-x, The number of oranges left with both Abdul and Sneha are x —3 and y —3 respectively., The product of number of left oranges = 120, , , , , , > (x-3)(y-3, > (x-3) G0-x-3 (2 y=30-x), > (x3) (27—x) = 120, > 27x —x° — 81 + 3x = 120 > x” 30x + 201=0, (b) Let the breadth of the plot be x. Then the length of the plot = 2x + 2, Since, area of the plot = 428 m (Given), s x(2x + 2) = 428, — 2x? + 2-428 =0, > xv +x-214=0, , Roots of a Quadratic Equation, , A real number « is called a root of the quadratic equation ax + bxt+c= 0,a40if ac? +batc=0., In other words, x = a is a root or solution of the quadratic equation ax’ + bx + c= 0 as it satisfies it., Example 3. Determine whether 3 is a root of the equation, , Vx? 4x43 +x? -9 = V4? - 1404 6