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Mathematics Chapter 4

Quadratic Equations

Exercise 4.1, 4.2, 4.3, 4.4 - step by step solutions

Exercise 4.1 - Verify and solve by factorization

Q1. x^2 - 3x - 10 = 0v

Step 1: Product = -10, sum = -3 -> -5 and +2.

Step 2: x^2 -5x +2x -10 = 0.

Step 3: x(x-5)+2(x-5)=0 => (x-5)(x+2)=0.

Final: x = 5, -2.

Q2. 2x^2 + x - 6 = 0v

Step 1: Product 2*(-6)=-12, sum=1 -> 4 and -3.

Step 2: 2x^2 +4x -3x -6=0.

Step 3: 2x(x+2)-3(x+2)=0 => (x+2)(2x-3)=0.

Final: x = -2, 3/2.

Q3. x^2 + 7x + 12 = 0v

Step 1: Product 12, sum 7 -> 3 and 4.

Step 2: (x+3)(x+4)=0.

Final: x = -3, -4.

Q4. 3x^2 - 8x + 4 = 0v

Step 1: Product 3*4=12, sum -8 -> -6 and -2.

Step 2: 3x^2 -6x -2x +4=0.

Step 3: 3x(x-2)-2(x-2)=0 => (x-2)(3x-2)=0.

Final: x = 2, 2/3.

Q5. x^2 - 4x + 4 = 0v

Step 1: x^2 -4x +4 = (x-2)^2.

Step 2: (x-2)^2=0 => x-2=0.

Final: x = 2 (double root).

Q6. 6x^2 - x - 2 = 0v

Step 1: Product 6*(-2)=-12, sum -1 -> -4 and +3.

Step 2: 6x^2 -4x +3x -2=0.

Step 3: 2x(3x-2)+1(3x-2)=0 => (3x-2)(2x+1)=0.

Final: x = 2/3, -1/2.

Exercise 4.2 - Completing the square

Q1. x^2 + 6x - 7 = 0v

x^2+6x=7 -> x^2+6x+9=16 -> (x+3)^2=16.

x+3=+/-4.

Final: x=1, -7.

Q2. x^2 - 10x + 9 = 0v

x^2-10x=-9 -> x^2-10x+25=16 -> (x-5)^2=16.

x-5=+/-4.

Final: x=9, 1.

Q3. 2x^2 + 4x - 3 = 0v

2x^2+4x=3 -> x^2+2x=3/2.

x^2+2x+1=5/2 -> (x+1)^2=5/2.

x+1=+/-sqrt(10)/2.

Final: x=-1 +/- sqrt(10)/2.

Q4. x^2 - 2x - 8 = 0v

x^2-2x=8 -> x^2-2x+1=9 -> (x-1)^2=9.

Final: x=4, -2.

Exercise 4.3 - Quadratic formula

Q1. 2x^2 - 4x - 6 = 0v

a=2, b=-4, c=-6.

x=[-b +/- sqrt(b^2-4ac)]/2a = [4 +/- sqrt(16+48)]/4 = [4 +/- 8]/4.

Final: x=3, -1.

Q2. x^2 + x - 1 = 0v

a=1, b=1, c=-1.

D=1+4=5.

x=(-1 +/- sqrt5)/2.

Final: x=(-1+sqrt5)/2, (-1-sqrt5)/2.

Q3. 3x^2 - x + 1 = 0v

a=3, b=-1, c=1.

D=b^2-4ac=1-12=-11.

Final: D<0, so no real roots.

Q4. 4x^2 + 4x + 1 = 0v

a=4, b=4, c=1.

D=16-16=0.

x=-b/2a=-4/8=-1/2.

Final: x=-1/2 (repeated).

Exercise 4.4 - Word problems (step by step)

Q1. Product of two consecutive positive integers is 306. Find the integers.v

Let numbers be x and x+1.

x(x+1)=306 -> x^2+x-306=0.

(x+18)(x-17)=0 -> x=17 or -18.

Positive integer taken.

Final: 17 and 18.

Q2. Train covers 360 km. If speed increases by 5 km/h, time reduces by 1 h. Find speed.v

Let speed=x km/h.

360/x - 360/(x+5)=1.

360(x+5)-360x=x(x+5).

1800=x^2+5x -> x^2+5x-1800=0.

(x+45)(x-40)=0.

Final: speed = 40 km/h.

Q3. Rectangular plot area 528 m^2, length is 1 more than twice breadth. Find dimensions.v

Let breadth=x, length=2x+1.

x(2x+1)=528 -> 2x^2+x-528=0.

D=1+4224=4225=65^2.

x=[-1+65]/4=16 (positive), so breadth=16.

Length=2*16+1=33.

Final: breadth 16 m, length 33 m.

Q4. Sum of ages of two friends is 20 years. Product is 96. Find ages.v

Let ages be x and (20-x).

x(20-x)=96 -> x^2-20x+96=0.

(x-12)(x-8)=0.

Final: ages are 12 years and 8 years.

Chapter 4 corrected with detailed steps.

Now continuing Chapter 5 onwards.

Next Chapter: Arithmetic Progressions ->

NCERT Class 10 Maths - Chapter 4