Implemented Components

Check whether the following are quadratic equations :

(i)x+12=2x−3

First expand both sides: x+12=2+2x+1 2(x−3)=2x−6

Now equate both sides: x2+2x+1=2x−6

Simplifying the eqn, we get:

This equation can be written in the standard quadratic form ax2+bx+c=0, where: a=1,b=0,c=7 this is a equation

(ii)x2−2x=−23−x

First expand the right side:

The equation becomes: x2−2x=−6+2x

Simplifying the eqn, we get:

This equation can be written in the standard quadratic form , where: 𝑎=1,𝑏=-4,𝑐=6

(iii) x−2x+1=x−1x+3**

First expand both sides: x−2x+1=x2+x−2x−2 x−1x+3=x2+3x−x−3=x2+2x−3

Now equate both sides: x2−x−2=x2+2x−3

Simplifying the eqn, we get:−3x+1=0

This equation can be written in the standard quadratic form , where: 𝑎=0,𝑏=-3,𝑐=1

(iv)x−32x+1=xx+5**

First expand both sides:

Now equate both sides: 2x2−5x−3=x2+5x

Simplifying the eqn, we get:x2−10x−3=0

This equation can be written in the standard quadratic form , where:𝑎=1,𝑏=-10,𝑐=-3

(v)2x−1x−3=x+5x−1

First expand both sides: 2x−1x−3=2xx−3−1x−3=2x2−6x−x+3=2x2−7x+3 x+5x−1=xx−1+5x−1=x2−x+5x−5=x2+4x−5

Now equate both sides: 2x2−7x+3=x2+4x−5

Simplifying the eqn, we get:x2−11x+8=0

This equation can be written in the standard quadratic form , where: 𝑎=1 𝑏=-11 𝑐=8

(vi)x2+3x+1=x−22

First expand right sides: x−22=x2+4−2x

Now equate both sides:x2+3x+1=x2+4−2x

Simplifying the eqn, we get:

This equation can be written in the standard quadratic form , where: 𝑎=0 𝑏=5 𝑐=-3

(vii)x+23=2xx2−1

First expand both sides: x+23=x3+8+6x2+12x=x3+6x2+12x+8 2x3−2x

Now equate both sides: x3+6x2+12x+8=2x3−2x

Simplifying the eqn, we get: x3−6x2−14x−8=0

This equation can be written in the standard quadratic form , where: 𝑎=1 𝑏=-6 𝑐=-14 d= -8

(viii)x3−4x2−x+1=x−23

First expand right side: x−23=x3−8−6x2+12x=x3−6x2+12x−8

Now equate both sides: x3−4x2−x+1=x3−6x2+12x−8

Simplifying the eqn, we get: 2x2−13x+9=0

This equation can be written in the standard quadratic form , where: 𝑎=2 𝑏=-13 𝑐=9

1. The area of a rectangular plot is 528 m2 . The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

find the length and breadth of the plot.

Now, Let x be the breadth of the rectangular plot in meters.
The length y is given to be one more than twice the breadth (Step 1)
Area=length×breadth (Step 2)
Substituting given values (Step 3)
Now, simplify and expand the equation: (Step 4)
Rearranging the equation into standard form
The equation is 2b2+b−528=0

2. The product of two consecutive positive integers is 306. We need to find the integers

(ii) We need to find the integers.

Now, Let one integer be x.
The other integer would be x+1.
Writing the equation(Step 1)
Simplifying the equation (Step 2)
Rearranging the equation into standard form (Step 3)
The equation is x2+x−306=0

3.Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.

(iii) find Rohan’s present age.

Now, Let Rohan’s current age be x years.
Rohan’s mother’s current age will be x+26 years.
In 3 years, Rohan will be x+3 years and his mother will be x+29 years.
Writing the equation (Step 1)
Now, simplify and expand the equation: (Step 2)
Combine like terms (Step 3)
Rearranging the equation into standard form (Step 4)
The equation is x2+32x−273=0

4.A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

find the speed of the train.

Now, Let initial speed be x.
Speed if it had been 8 kmph less would be (x-8). (Step 1)
Writing the equation (Step 2)
Clear the fractions by multiplying through by 𝑥(𝑥−8): (Step 3)
Simplify the equation: (Step 4)
Combine like terms and rearrange the equation into standard form (Step 4)
The equation is x2−8x−1280=0

(i)x2−3x−10=0

To find roots of the given equation:
Identify the coefficients:
a=1 (coefficient of x2)
𝑏=−3(coefficient of x)
𝑐=−10(constant term)
Find two numbers that multiply to 𝑎𝑐 and add to 𝑏: (Step 2)
Rewrite the middle term using these numbers: (Step 3)
Factor by grouping:: (Step 4)
Factor out the common binomial factor:(Step 4)
Set each factor equal to zero to find the roots:
The roots are: ,

(ii) 2x2+x−6=0

To find roots of the given equation:
Identify the coefficients:
a=2 (coefficient of x2)
𝑏=1(coefficient of x)
𝑐=−6(constant term)
Find two numbers that multiply to 𝑎𝑐 and add to 𝑏: (Step 2)
Rewrite the middle term using these numbers: (Step 3)
Factor by grouping:: (Step 4)
Factor out the common binomial factor:(Step 5)
Set each factor equal to zero to find the roots:
The roots are: ,

(iii) 212·x2+7x+5·212

To find roots of the given equation:
Identify the coefficients:
a=2^1/2 (coefficient of x2)
𝑏=7(coefficient of x)
𝑐=5*(2^1/2)(constant term)
Find two numbers that multiply to 𝑎𝑐 and add to 𝑏: (Step 2)
Rewrite the middle term using these numbers: (Step 3)
Factor by grouping: (Step 4)
Factor out the common binomial factor:(Step 5)
Set each factor equal to zero to find the roots:
The roots are: ,

(iv) 2x2−x+18

To find roots of the given equation:
Identify the coefficients:
a=2(coefficient of x2)
𝑏=-1(coefficient of x)
𝑐=1/8(constant term)
Multply the whole equation by 8 and find two numbers that multiply to 𝑎𝑐 and add to 𝑏: (Step 2)
Rewrite the middle term using these numbers: (Step 3)
Factor by grouping: (Step 4)
Factor out the common binomial factor:(Step 5)
Set each factor equal to zero to find the roots:
The roots are: ,

(v) 100x2−20x+1=0

To find roots of the given equation:
Identify the coefficients:
a=100(coefficient of x2)
𝑏=-20(coefficient of x)
𝑐=1(constant term)
Rewrite the middle term using these numbers: (Step 3)
Factor by grouping: (Step 4)
Factor out the common binomial factor:(Step 5)
Set each factor equal to zero to find the roots:
The roots are: ,

(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.

find out how many marbles

Now, Let x be the no.of marbles John had.
The length is given to be no.of marbles Jivanti had.
No.of marbles John had left after he lost 5 marbles=x-5
No.of marbles left with Jivanti when she lost 5 marbles=40-x
the product of the no.of marbles they have now is =124
Rearranging the equation into standard form
The equation is x2−45x+324=0

ii.A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was ` 750. We would like to find out the number of toys produced on that day.

find out the number of toys.

Now, Let x be the no.of toys produced on that day.
The cost of production (in rupees) of each toy that day is .
Total cost of production that day= =750
Rearranging the equation into standard form
The equation is

(iii)Find two numbers whose sum is 27 and product is 182* Now, Let x be the no.of toys produced on that day.

Find two numbers whose sum is 27

Let the two numbers be x and y.
The sum is given by x+y=27
The product is given by xy=182
y=27−x
Substitute y in the product equation.
x27−x=182
27x−x2=182
The new equation is
To find roots of the given equation:
Identify the coefficients:
a=1(coefficient of x2)
𝑏=-27(coefficient of x)
𝑐=182(constant term)
Rewrite the middle term using these numbers: (Step 3)
Factor by grouping: (Step 4)
Factor out the common binomial factor:(Step 5)
Set each factor equal to zero to find the roots:
The roots are: ,

Find two consecutive positive integers, sum of whose squares is 365. The two integers are and .

5.** The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.** The other two sides are cm and cm.

**A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was 90,findthenumberofarticlesproducedandthecostofeacharticle.∗Thenumberofarticlesproducedis6andcostofeacharticleisrupee` .

To find the values of 𝑘k for the quadratic equation 2x^2 + kx + 3 = 0 so that it has two equal roots, we need to make its discriminant Delta equal to zero.

The discriminant (\Delta) of a quadratic equation ax^2 + bx + c = 0 is given by:

Solve

Delta = b^2 - 4ac
For the equation 2x2 + kx + 3 = 0:
a = 2 (coefficient of x2)
b = k (coefficient of x)
c = 3 (constant term)
Substitute these values into the discriminant formula:
Delta = k2 - 4(2)(3)
For the equation to have two equal roots, the discriminant must be equal to zero:
k^2 - 24 = 0
Now, solve for k:
k2 = 24
k = pm \sqrt{24}
k = pm 2\sqrt{6}
So, the values of k for the equation 2x^2 + kx + 3 = 0 to have two equal roots are k = 2\sqrt{6} and k = -2\sqrt{6}.

To determine if it is possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m², we need to solve the following problem:

solve

Let the breadth of the rectangle be b meters.
Since the length is twice the breadth, the length will be 2b meters.
The area of the rectangle is given by the product of its length and breadth.
Set up the equation for the area:
Area = length × breadth
800 = 2b × b
Simplify the equation:
800 = 2b^2
Divide both sides by 2:
400 = b^2
Solve for b by taking the square root of both sides:
b = sqrt(400)
b =
So, the breadth is 20 meters.
Since the length is twice the breadth, the length will be:
length = 2b
length = 2(20)
length =
So, the length is 40 meters.
Therefore, it is possible to design a rectangular mango grove with the given conditions. The dimensions are:
length = meters
breadth = meters

To determine if it is possible to design a rectangular park with a perimeter of 80 meters and an area of 400 square meters, we need to solve the following problem:

solve

Let the length of the rectangle be meters.

Let the breadth of the rectangle be b meters.
The perimeter of the rectangle is given by:
Perimeter = 2(l + b)
80 = 2(l + b)
Simplify the equation:
l + b = 40
The area of the rectangle is given by:
Area = l * b
400 = l * b
We now have a system of two equations to solve:
l + b = 40
l * b = 400
Express b in terms of l using the first equation:
b = 40 - l
Substitute b in the second equation:
l * (40 - l) = 400
40l - l2 = 400
Rewrite the equation as a standard quadratic equation:
l^2 - 40l + 400 = 0
Solve the quadratic equation using the quadratic formula l = −b±b2−4ac2a:
Identify the coefficients:
a = 1, b = -40, c = 400
Calculate the discriminant Delta:
Delta = b2 - 4ac
Delta = −402 - 4(1)(400)
Delta = 1600 - 1600
Delta =
Since the discriminant is zero (Delta = 0), the quadratic equation has exactly one real root:
l = −−40±02·1
l = 402
l =
Substitute l = back into the equation b = 40 - l to find b:
b = 40 - 20
b =
So, the length is meters and the breadth is meters.
Therefore, it is possible to design a rectangular park with the given conditions. The dimensions are:
length = 20 meters, breadth = 20 meters

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