Exercise 4.1 2 Questions – Check Quadratic Equations
Check whether the following are quadratic equations:
(i) (x+1)² = 2(xâˆ'3) â‡' x²+2x+1 = 2xâˆ'6 â‡' x²+7 = 0 â†' Yes, quadratic (can write as x²+0·x+7=0)
(ii) x²âˆ'2x = (âˆ'2)(3âˆ'x) â‡' x²âˆ'2x = âˆ'6+2x â‡' x²âˆ'4x+6 = 0 â†' Yes
(iii) (xâˆ'2)(x+1) = (xâˆ'1)(x+3) â‡' x²âˆ'xâˆ'2 = x²+2xâˆ'3 â‡' âˆ'3x+1 = 0 â†' No (linear)
(iv) (xâˆ'3)(2x+1) = x(x+5) â‡' 2x²âˆ'5xâˆ'3 = x²+5x â‡' x²âˆ'10xâˆ'3 = 0 â†' Yes
(v) (2xâˆ'1)(xâˆ'3) = (x+5)(xâˆ'1) â‡' 2x²âˆ'7x+3 = x²+4xâˆ'5 â‡' x²âˆ'11x+8 = 0 â†' Yes
(vi) x²+3x+1 = (xâˆ'2)² â‡' x²+3x+1 = x²âˆ'4x+4 â‡' 7xâˆ'3 = 0 â†' No
(vii–viii) Check similarly â†' expand, simplify, check if x² term remains with non-zero coefficient.
Represent the following situations in the form of quadratic equations:
(i) The area of a rectangular plot is 528 m². The length is 1 more than twice its breadth.
Let breadth = x. Length = 2x+1. Area = x(2x+1) = 528 â‡' 2x² + x âˆ' 528 = 0
(ii) The product of two consecutive positive integers is 306.
Let integers be x, x+1. x(x+1)=306 â‡' x² + x âˆ' 306 = 0
(iii) John and Jivanti together have 45 marbles. Both lost 5 each, product of their marbles is 124.
John = x. Jivanti = 45âˆ'x. (xâˆ'5)(40âˆ'x) = 124 â‡' x² âˆ' 45x + 324 = 0
Exercise 4.2 6 Questions – Factorisation Method
(i) x² âˆ' 3x âˆ' 10 = 0
x²âˆ'5x+2xâˆ'10 = x(xâˆ'5)+2(xâˆ'5) = (xâˆ'5)(x+2) = 0. ∴ x = 5, âˆ'2
(ii) 2x² + x âˆ' 6 = 0
2x²+4xâˆ'3xâˆ'6 = 2x(x+2)âˆ'3(x+2) = (x+2)(2xâˆ'3) = 0. ∴ x = âˆ'2, 3/2
(iii) √2x² + 7x + 5√2 = 0
√2x²+5x+2x+5√2 = x(√2x+5)+√2(√2x+5) = (√2x+5)(x+√2) = 0. ∴ x = âˆ'5/√2, âˆ'√2
(iv) 2x² âˆ' x + 1/8 = 0
Multiply by 8: 16x²âˆ'8x+1 = 0 â‡' (4xâˆ'1)² = 0. ∴ x = 1/4 (repeated)
(v) 100x² âˆ' 20x + 1 = 0
(10xâˆ'1)² = 0. ∴ x = 1/10 (repeated)
Q2: Find two numbers whose sum is 27 and product is 182. â†' x(27âˆ'x)=182 â†' x²âˆ'27x+182=0 â†' 13 and 14
Q3: Find two consecutive positive integers sum of whose squares is 365. â†' x²+(x+1)²=365 â†' 13 and 14
Q4: Altitude of right triangle is 7 cm less than base. Hypotenuse = 13 cm. Find the other two sides. â†' 5 cm, 12 cm
Q5: A cottage industry produces pottery articles daily. Cost of each article is twice the number of articles plus 3. If total cost is ₹90, find articles produced and cost per article. â†' 6 articles, ₹15 each
Q6: Find dimensions of a rectangle whose perimeter is 28 m and diagonal is 10 m. â†' 6 m × 8 m
Exercise 4.3 11 Questions – Completing the Square & Quadratic Formula
Q1: Find roots by completing square: 2x²âˆ'7x+3=0
Divide by 2: x²âˆ'(7/2)x+3/2=0. x²âˆ'(7/2)x=âˆ'3/2. Add (7/4)²=49/16 to both sides: (xâˆ'7/4)²=49/16âˆ'24/16=25/16. xâˆ'7/4=±5/4. x=3, 1/2
Quadratic Formula: x = [âˆ'b ± √(b²âˆ'4ac)] / 2a
Q2: Solve 2x²+xâˆ'4=0 using formula: x = [âˆ'1±√(1+32)]/4 = [âˆ'1±√33]/4
Q3: 4x²+4√3x+3=0 â†' (2x+√3)²=0 â†' x=âˆ'√3/2
Q5: In a class test, sum of Shefali's marks = 30. If she got 2 more in Maths and 3 less in English, product would be 210. Find marks. â†' Eng=13, Maths=17 OR Eng=12, Maths=18
Q6: Diagonal of rectangular field is 60 m more than shorter side. Longer side is 30 m more than shorter side. Find sides. â†' x²+(x+30)²=(x+60)² â†' 90 m, 120 m
Q8: A train travels 360 km at uniform speed. If speed is 5 km/h more, time reduces by 1 hr. Find speed. â†' 360/x âˆ' 360/(x+5) = 1 â†' 40 km/h
Q9: Two water taps fill a tank in 9â…œ hrs. Larger tap takes 10 hrs less than smaller. Find time for each. â†' 25 hrs and 15 hrs
Q11: Sum of areas of two squares = 468 m². Perimeter difference = 24 m. Find sides. â†' 12 m, 18 m
Exercise 4.4 5 Questions – Nature of Roots
Find the nature of roots: D = b²âˆ'4ac. If D>0: two distinct real roots. D=0: two equal real roots. D<0: no real roots.
(i) 2x²âˆ'3x+5=0 â†' D=9âˆ'40=âˆ'31<0 â†' No real roots
(ii) 3x²âˆ'4√3x+4=0 â†' D=48âˆ'48=0 â†' Two equal real roots
(iii) 2x²âˆ'6x+3=0 â†' D=36âˆ'24=12>0 â†' Two distinct real roots
Q2: kx(xâˆ'2)+6=0 has equal roots. Find k. â†' kx²âˆ'2kx+6=0. D=4k²âˆ'24k=0 â†' k = 6
Q3: Is it possible to design a rectangular mango grove with length = 2×breadth and area = 800 m²? â†' x(2x)=800, x²=400, x=20. Yes, 20×40 m
Q4: Sum of ages of two friends is 20 years. Four years ago, product was 48. Find ages. â†' 16 and 4 (not possible both) â†' Not possible
Q5: Perimeter 80 m, area 400 m². Possible? â†' x+y=40, xy=400. x(40âˆ'x)=400, x²âˆ'40x+400=0, D=0. Yes, square 20×20 m
Practice This Chapter
Generate unlimited practice questions for Quadratic Equations with Examzo — AI-powered, faculty-validated, zero hallucinations.
Try PaperGenie → Take a Quick Test →