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Note on Quadratic equation

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An equation like ax2+ bx + c = 0 where a ≠ 0, which contains only one variable and '2' as its highest power is called a quadratic equation. There are two values of the variable in any quadratic equation. The roots of the equations are :

$$ x = \frac {-b \pm \sqrt {b^2 -4ac}}{2a} $$

.

Quadratic equation are of two types

i. Pure quadratic equation

e.g. x2 = 9 or, x2 - 9 = 0
i.e. ax2 + c = 0 ( a ≠ 0, c = 0 )

ii. Adfected quadratic equation

e.g. x2 - 9x - 15 = 0
i.e. ax2 + bx + c = 0 ( a ≠ 0, b ≠ 0)

An equation like ax2+ bx + c = 0 where a≠ 0, which contains only one variable and '2' as its highest power is called a quadratic equation. There are two values of the variable in any quadratic equation.. The roots of the equations are :

$$ x = \frac {-b \pm \sqrt {b^2 -4ac}}{2a} $$

.

Very Short Questions

Solution:

4x2 = 16x

or,4x2- 16x = 0

or, 4x ( x - 4) = 0

Either 4x = 0

∴ x = 0

Or, x - 4 = 0

∴ x = 4

Solution:

\( \frac{x^2+3}{2}\) = 6

or, \( x^ 2 + 3 \) = 12

or, \( x^2\) = 9

or, \((x)^2\) = (±3)2

or, x =±3

Let, the number be x. Hence, the other number is \( x^2\)

According to question,

x + \( x^2\) = 20

or,x + \( x^2\) - 20 = 0

or, \(x^2\) + 5x - 4x - 20 = 0

or, x ( x + 5) - 4(x + 5) = 0

or, (x - 4) (x + 5) = 0

Either, x - 4 = 0

∴ x = 4


Or, x + 5 = 0

∴ x = -5

Let, the three numbers be x, x + 1 and x + 2

Then, According to question,

\(x ^2\) + \( (x+1)^2\) + \( (x+2)^2\) = 194

or, \(x ^2\) + \(x ^2\) + 2x + 1 + \(x ^2\) + 4x + 4 = 194

or, 3\(x ^2\) + 6x - 189 = 0

or, 3(\(x ^2\) + 2x - 63) = 0

or, (\(x ^2\) + 2x - 63) = 0

or, \( x^2\) + 9x - 7x - 63 = 0

or, x (x + 9) - 7( x+9) = 0

or, ( x - 7) ( x + 9) = 0

Either,

x - 7 = 0

∴ x = 7

Or, x + 9 = 0

∴ x = -9.

Therefore, the required numbers are 7, 7 + 1 , 7 + 2 = 7, 8, 9

Or, -9, -9 + 1, -9 + 2 = -9, -8, -7

Let, one of the odd number be x . Then, its consecutive odd number is ( x + 2)

According to question,

x ( x + 2) = 143

or, \( x^2\) + 2x = 143

or, \( x ^ 2\) + 2x - 143 = 0

or, \( x ^2\) + 13x - 11x - 143 = 0

or, x ( x + 13) - 11( x + 13) = 0

or, ( x + 13) ( x - 11) = 0

Either,

x - 11 = 0

or, x = 11

Or, x + 13 =0

or, x = -13

Hence, the required number is 11 and 11 + 2 = 13

Let, one of the evennumber be x . Then, its consecutive evennumber is ( x + 2)

According to question,

x ( x + 2) = 168

or, \( x^2\) + 2x = 168

or, \( x ^ 2\) + 2x - 168 = 0

or, \( x ^2\) + 14x - 12x - 168 = 0

or, x ( x + 14) - 12( x + 14) = 0

or, ( x + 14) ( x - 12) = 0

Either,

x - 12 = 0

or, x = 12

Or, x + 14 = 0

or, x = -14

Hence, the required number is 12 and 11 + 2 = 14

Let, one of the number be x . Then, its consecutive number is ( x + 1)

According to question,

x ( x + 1) =56

or, \( x^2\) + x =56

or, \( x ^ 2\) + x - 56= 0

or, \( x ^2\) + 8x - 7x - 56= 0

or, x ( x + 8) - 7( x + 8) = 0

or, ( x + 8) ( x - 7) = 0

Either,

x - 7= 0

or, x =7

Or, x + 8=0

or, x = -8

Hence, the required number is 7 and 7 + 1=8

Let, the odd number be x and its consecutive odd number be ( x + 2).

According to question,

\( (x + 2) ^2\) - \( x^2\) = 24

or,\( x^2\) + 4x + 4 -\( x^2\) = 24

or, 4x = 24 - 4

or, 4x = 20

or, x = \( \frac {20}{4}\)

or, x = 5

Hence, the required numbers are 5 and 5 + 2 = 7

Let, the number be x.

Then, According to question,

4\( x ^ 2\) = 16x

or 4 \( x ^2\) - 16x = 0

or, 4x (x - 4) = 0

Either, 4x = 0

∴ x = 0

Or, x - 4 = 0

∴ x = 4.,

Let, the number be x.

Then, According to question,

\( x ^ 2\) - 9 = 40

or,\( x ^ 2\) = 40 + 9

or,\( x ^ 2\) = 49

or, \( (x)^2\) = \( (±7) ^ 2\)

∴x =±7

Let the whole number be x

Then, According to question,

 

x - 10 = 39× \( \frac {1}{x}\)

or, \( x ^ 2 \) - 10x = 39

or,\( x ^ 2 \) - 10x - 39 = 0

or,\( x ^ 2 \) - 13x + 3x - 39 = 0

or, x ( x - 13) + 3(x - 13) = 0

( x + 3) ( x - 13) = 0

Either x - 13 = 0

∴ x = 13

Or, x + 3 = 0

∴ x = - 3

 

Hence, the required whole number is 13.

Let, the natural number be x

Then,

According to quesiton,

\( \frac {x ^2}{2}\) - 5 = 45

or \( \frac {x ^2}{2}\) = 45 + 5

or,\( \frac {x ^2}{2}\)= 50

or, \( x ^ 2\) = 100

or, \( ( x)^2\) = \( (10)^2\) [ Only (+10) is taken because the required number is a natural number]

or, x = 10

Hence, the requierd number is 10.

 

Let, the first number is x then the second number becomes (12 - x)

According to question,

x ( 12 - x) = 32

or, -\( x ^ 2\) + 12x - 32 = 0

or, \( x ^ 2\) - 12x + 32= 0

or, \( x ^ 2\) - 8x - 4x + 32 = 0

or, x ( x - 8) - 4( x - 8 ) = 0

or, ( x - 4) ( x - 8) = 0

Either, x - 4 = 0

∴ x = 4

Or, x - 8 = 0

∴ x = 8

If first number is 4, the other number is 12 - 4 = 8

If first number is 8, the second number is 12 - 8 = 4

Hence, the required numbers are 8,4 or 4,8

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  • If the sum of two numbers is 10 and their product is 24, then what are the numbers?

    6,4
    6,7
    6,6
    6,5
  • The sum of digits of two digit number is 7 and their product is 12. What are the numbers?

    43,34
    34,35
    43,36
    43,33
  • The sum of two numbers is 16 and the sum of their squares is 130. Find the numbers.

    9 and 8
    9 and 7
    9 and 9
    none
  • If the sum of two numbers is  9 and their product is 18, then what are the numbers?

    6 and 3
    6 and 5
    6 and 2
    6 and 4
  • Find two consecutive even number whose product is 48.  

    6,8
    6,7
    all
    6,6
  • The difference between the age of two sisters is 5 years and the product is 204. What is the age of the two sisters?

    none
    12,18
    12,17
    13,17
  • Sujit is 7 years older than Amisha. two years ago, the product of their ages was 18. What is their present age?

    11,3
    11,5
    all
    11,4
  • A perimeter of a rectangular ground is 46 m and its area is 126 sq, meters. What is the length and breadth of the ground?

    15m, 9m
    14m , 9m
    14m, 8m
    15m, 9m
  • The hypotenuse of a right-angled triangle is 15 cm. If the ratio of the remaining two sides is 3:4, find the two sides.

    9 cm, 12 cm
    9 cm , 11 cm
    13cm, 9cm
    11 cm, 9cm
  • Rs.120 is equally divided among a certain number of students. If there were 3 students more, each would have received Rs. 2 less. What is the numbers of students?

    12
    13
    15
    14
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DISCUSSIONS ABOUT THIS NOTE

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If the age of father was 13 times the age of his son before 1 year.Now the age of fatherbis cube of his son age.find their present aged.

Solve please


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Gyan

If two times the number exceeds,the number formed by interchanging its digit by four ???what does it mean ??


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Hiranya

Ask any queries on this note.x^2-4x 4=0


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Ask any queries on this notex^2_4X 4=0


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