RS Aggarwal Solutions: Quadratic Equations (Exercise 4E)

``` Page 1

1. The sum of a natural number and its square is 156. Find the number.
Sol:
Let the required natural number be x.
According to the given condition,
2
2
2
156
156 0
13 12 156 0
13 12 13 0
13 12 0
13 0 12 0
13 12
x x
x x
x x x
x x x
x x
x or x
x or x
12 x ( x cannot be negative)
Hence, the required natural number is 12.
2. The sum of natural number and its positive square root is 132. Find the number.
Sol:
Let the required natural number be x.
According to the given condition,
132 x x
Putting
2
, x y o r x y we get
2
2
2
132
132 0
12 11 132 0
12 11 12 0
12 11 0
12 0 11 0
12 11
y y
y y
y y y
y y y
y y
y or y
y or y
11 y ( y cannot be negative)
Now,
2
11
11 121
x
x
Hence, the required natural number is 121.
Page 2

1. The sum of a natural number and its square is 156. Find the number.
Sol:
Let the required natural number be x.
According to the given condition,
2
2
2
156
156 0
13 12 156 0
13 12 13 0
13 12 0
13 0 12 0
13 12
x x
x x
x x x
x x x
x x
x or x
x or x
12 x ( x cannot be negative)
Hence, the required natural number is 12.
2. The sum of natural number and its positive square root is 132. Find the number.
Sol:
Let the required natural number be x.
According to the given condition,
132 x x
Putting
2
, x y o r x y we get
2
2
2
132
132 0
12 11 132 0
12 11 12 0
12 11 0
12 0 11 0
12 11
y y
y y
y y y
y y y
y y
y or y
y or y
11 y ( y cannot be negative)
Now,
2
11
11 121
x
x
Hence, the required natural number is 121.

3. The sum of two natural number is 28 and their product is 192. Find the numbers.
Sol:
Let the required number be x and 28 . x
According to the given condition,
2
2
2
28 192
28 192
28 192 0
16 12 192 0
16 12 16 0
12 16 0
12 0 16 0
x x
x x
x x
x x x
x x x
x x
x or x
12 16 x o r x
When 12, x
28 28 12 16 x
When 16, x
28 28 16 12 x
Hence, the required numbers are 12 and 16.
4. The sum of the squares of two consecutive positive integers is 365. Find the integers.
Sol:
Let the required two consecutive positive integers be x and 1 . x
According to the given condition,
2
2
2 2
2
2
2
1 365
2 1 365
2 2 364 0
182 0
14 13 182 0
x x
x x x
x x
x x
x x x
14 13 14 0
14 13 0
14 0 13 0
14 13
x x x
x x
x or x
x or x
13 x ( x is a positive integers)
When 13, x
1 13 1 14 x
Page 3

1. The sum of a natural number and its square is 156. Find the number.
Sol:
Let the required natural number be x.
According to the given condition,
2
2
2
156
156 0
13 12 156 0
13 12 13 0
13 12 0
13 0 12 0
13 12
x x
x x
x x x
x x x
x x
x or x
x or x
12 x ( x cannot be negative)
Hence, the required natural number is 12.
2. The sum of natural number and its positive square root is 132. Find the number.
Sol:
Let the required natural number be x.
According to the given condition,
132 x x
Putting
2
, x y o r x y we get
2
2
2
132
132 0
12 11 132 0
12 11 12 0
12 11 0
12 0 11 0
12 11
y y
y y
y y y
y y y
y y
y or y
y or y
11 y ( y cannot be negative)
Now,
2
11
11 121
x
x
Hence, the required natural number is 121.

3. The sum of two natural number is 28 and their product is 192. Find the numbers.
Sol:
Let the required number be x and 28 . x
According to the given condition,
2
2
2
28 192
28 192
28 192 0
16 12 192 0
16 12 16 0
12 16 0
12 0 16 0
x x
x x
x x
x x x
x x x
x x
x or x
12 16 x o r x
When 12, x
28 28 12 16 x
When 16, x
28 28 16 12 x
Hence, the required numbers are 12 and 16.
4. The sum of the squares of two consecutive positive integers is 365. Find the integers.
Sol:
Let the required two consecutive positive integers be x and 1 . x
According to the given condition,
2
2
2 2
2
2
2
1 365
2 1 365
2 2 364 0
182 0
14 13 182 0
x x
x x x
x x
x x
x x x
14 13 14 0
14 13 0
14 0 13 0
14 13
x x x
x x
x or x
x or x
13 x ( x is a positive integers)
When 13, x
1 13 1 14 x

Hence, the required positive integers are 13 and 14.
5. The sum of the squares to two consecutive positive odd numbers is 514. Find the numbers.
Sol:
Let the two consecutive positive odd numbers be x and 2 . x
According to the given condition,
2
2
2 2
2
2
2
2 514
4 4 514
2 4 510 0
2 255 0
17 15 255 0
17 15 17 0
17 15 0
17 0 15 0
17 15
x x
x x x
x x
x x
x x x
x x x
x x
x or x
x or x
15 x ( x is a positive odd number)
When 15, x
2 15 2 17 x
Hence, the required positive integers are 15 and 17.
6. The sum of the squares of two consecutive positive even numbers is 452. Find the numbers.
Sol:
Let the two consecutive positive even numbers be x and 2 . x
According to the given condition,
2
2
2 452 x x
2 2
4 4 452 x x x
2
2 4 448 0 x x
2
2 224 0 x x
2
16 14 224 0 x x x
16 14 16 0
16 14 0
16 0 14 0
16 14
x x x
x x
x o r x
x o r x
14 x ( x is a positive even number)
When 14, x
2 14 2 16 x
Page 4

1. The sum of a natural number and its square is 156. Find the number.
Sol:
Let the required natural number be x.
According to the given condition,
2
2
2
156
156 0
13 12 156 0
13 12 13 0
13 12 0
13 0 12 0
13 12
x x
x x
x x x
x x x
x x
x or x
x or x
12 x ( x cannot be negative)
Hence, the required natural number is 12.
2. The sum of natural number and its positive square root is 132. Find the number.
Sol:
Let the required natural number be x.
According to the given condition,
132 x x
Putting
2
, x y o r x y we get
2
2
2
132
132 0
12 11 132 0
12 11 12 0
12 11 0
12 0 11 0
12 11
y y
y y
y y y
y y y
y y
y or y
y or y
11 y ( y cannot be negative)
Now,
2
11
11 121
x
x
Hence, the required natural number is 121.

3. The sum of two natural number is 28 and their product is 192. Find the numbers.
Sol:
Let the required number be x and 28 . x
According to the given condition,
2
2
2
28 192
28 192
28 192 0
16 12 192 0
16 12 16 0
12 16 0
12 0 16 0
x x
x x
x x
x x x
x x x
x x
x or x
12 16 x o r x
When 12, x
28 28 12 16 x
When 16, x
28 28 16 12 x
Hence, the required numbers are 12 and 16.
4. The sum of the squares of two consecutive positive integers is 365. Find the integers.
Sol:
Let the required two consecutive positive integers be x and 1 . x
According to the given condition,
2
2
2 2
2
2
2
1 365
2 1 365
2 2 364 0
182 0
14 13 182 0
x x
x x x
x x
x x
x x x
14 13 14 0
14 13 0
14 0 13 0
14 13
x x x
x x
x or x
x or x
13 x ( x is a positive integers)
When 13, x
1 13 1 14 x

Hence, the required positive integers are 13 and 14.
5. The sum of the squares to two consecutive positive odd numbers is 514. Find the numbers.
Sol:
Let the two consecutive positive odd numbers be x and 2 . x
According to the given condition,
2
2
2 2
2
2
2
2 514
4 4 514
2 4 510 0
2 255 0
17 15 255 0
17 15 17 0
17 15 0
17 0 15 0
17 15
x x
x x x
x x
x x
x x x
x x x
x x
x or x
x or x
15 x ( x is a positive odd number)
When 15, x
2 15 2 17 x
Hence, the required positive integers are 15 and 17.
6. The sum of the squares of two consecutive positive even numbers is 452. Find the numbers.
Sol:
Let the two consecutive positive even numbers be x and 2 . x
According to the given condition,
2
2
2 452 x x
2 2
4 4 452 x x x
2
2 4 448 0 x x
2
2 224 0 x x
2
16 14 224 0 x x x
16 14 16 0
16 14 0
16 0 14 0
16 14
x x x
x x
x o r x
x o r x
14 x ( x is a positive even number)
When 14, x
2 14 2 16 x

Hence, the required numbers are 14 and 16.
7. The product of two consecutive positive integers is 306. Find the integers.
Sol:
Let the two consecutive positive integers be x and 1 . x
According to the given condition,
2
2
1 306
306 0
18 17 306 0
18 17 18 0
18 17 0
18 0 17 0
18 17
x x
x x
x x x
x x x
x x
x or x
x or x
17 x ( x is a positive integers)
When 17, x
1 17 1 18 x
Hence, the required integers are 17 and 18.
8. Two natural number differ by 3 and their product is 504. Find the numbers.
Sol:
Let the required numbers be x and 3 . x
According to the question:
3 504 x x
2
3 504 x x
2
3 504 0 x x
2
24 21 504 0 x x
2
24 21 504 0 x x x
24 21 24 0 x x x
24 21 0
24 0 21 0
24 21
x x
x o r x
x o r x
If 24, x the numbers are 24 and 24 3 21 .
If 21, x the numbers are 21 and 21 3 24 .
Hence, the numbers are 24, 21 and 21,24 .
Page 5

1. The sum of a natural number and its square is 156. Find the number.
Sol:
Let the required natural number be x.
According to the given condition,
2
2
2
156
156 0
13 12 156 0
13 12 13 0
13 12 0
13 0 12 0
13 12
x x
x x
x x x
x x x
x x
x or x
x or x
12 x ( x cannot be negative)
Hence, the required natural number is 12.
2. The sum of natural number and its positive square root is 132. Find the number.
Sol:
Let the required natural number be x.
According to the given condition,
132 x x
Putting
2
, x y o r x y we get
2
2
2
132
132 0
12 11 132 0
12 11 12 0
12 11 0
12 0 11 0
12 11
y y
y y
y y y
y y y
y y
y or y
y or y
11 y ( y cannot be negative)
Now,
2
11
11 121
x
x
Hence, the required natural number is 121.

3. The sum of two natural number is 28 and their product is 192. Find the numbers.
Sol:
Let the required number be x and 28 . x
According to the given condition,
2
2
2
28 192
28 192
28 192 0
16 12 192 0
16 12 16 0
12 16 0
12 0 16 0
x x
x x
x x
x x x
x x x
x x
x or x
12 16 x o r x
When 12, x
28 28 12 16 x
When 16, x
28 28 16 12 x
Hence, the required numbers are 12 and 16.
4. The sum of the squares of two consecutive positive integers is 365. Find the integers.
Sol:
Let the required two consecutive positive integers be x and 1 . x
According to the given condition,
2
2
2 2
2
2
2
1 365
2 1 365
2 2 364 0
182 0
14 13 182 0
x x
x x x
x x
x x
x x x
14 13 14 0
14 13 0
14 0 13 0
14 13
x x x
x x
x or x
x or x
13 x ( x is a positive integers)
When 13, x
1 13 1 14 x

Hence, the required positive integers are 13 and 14.
5. The sum of the squares to two consecutive positive odd numbers is 514. Find the numbers.
Sol:
Let the two consecutive positive odd numbers be x and 2 . x
According to the given condition,
2
2
2 2
2
2
2
2 514
4 4 514
2 4 510 0
2 255 0
17 15 255 0
17 15 17 0
17 15 0
17 0 15 0
17 15
x x
x x x
x x
x x
x x x
x x x
x x
x or x
x or x
15 x ( x is a positive odd number)
When 15, x
2 15 2 17 x
Hence, the required positive integers are 15 and 17.
6. The sum of the squares of two consecutive positive even numbers is 452. Find the numbers.
Sol:
Let the two consecutive positive even numbers be x and 2 . x
According to the given condition,
2
2
2 452 x x
2 2
4 4 452 x x x
2
2 4 448 0 x x
2
2 224 0 x x
2
16 14 224 0 x x x
16 14 16 0
16 14 0
16 0 14 0
16 14
x x x
x x
x o r x
x o r x
14 x ( x is a positive even number)
When 14, x
2 14 2 16 x

Hence, the required numbers are 14 and 16.
7. The product of two consecutive positive integers is 306. Find the integers.
Sol:
Let the two consecutive positive integers be x and 1 . x
According to the given condition,
2
2
1 306
306 0
18 17 306 0
18 17 18 0
18 17 0
18 0 17 0
18 17
x x
x x
x x x
x x x
x x
x or x
x or x
17 x ( x is a positive integers)
When 17, x
1 17 1 18 x
Hence, the required integers are 17 and 18.
8. Two natural number differ by 3 and their product is 504. Find the numbers.
Sol:
Let the required numbers be x and 3 . x
According to the question:
3 504 x x
2
3 504 x x
2
3 504 0 x x
2
24 21 504 0 x x
2
24 21 504 0 x x x
24 21 24 0 x x x
24 21 0
24 0 21 0
24 21
x x
x o r x
x o r x
If 24, x the numbers are 24 and 24 3 21 .
If 21, x the numbers are 21 and 21 3 24 .
Hence, the numbers are 24, 21 and 21,24 .

9. Find two consecutive multiples of 3 whose product is 648.
Sol:
Le the required consecutive multiples of 3 be 3 x and 3 1 . x
According to the given condition,
2
2
2
2
3 3 1 648
9 648
72
72 0
9 8 72 0
9 8 9 0
9 8 0
9 0 8 0
9 8
x x
x x
x x
x x
x x x
x x x
x x
x or x
x or x
8 x (Neglecting the negative value)
When 8, x
3 3 8 24
3 1 3 8 1 3 9 27
x
x
Hence, the required multiples are 24 and 27.
10. Find the tow consecutive positive odd integer whose product s 483.
Sol:
Let the two consecutive positive odd integers be x and 2 . x
According to the given condition,
2 483 x x
2
2 483 0 x x
2
23 21 483 0 x x x
23 21 23 0 x x x
23 21 0
23 0 21 0
23 21
x x
x or x
x o r x
21 x ( x is a positive odd integer)
When 21, x
2 21 2 23 x
Hence, the required integers are 21 and 23.
```

## Mathematics (Maths) Class 10

120 videos|463 docs|105 tests

## FAQs on RS Aggarwal Solutions: Quadratic Equations (Exercise 4E) - Mathematics (Maths) Class 10

 1. How can I solve quadratic equations?
Ans. To solve quadratic equations, you can use various methods such as factoring, completing the square, or using the quadratic formula. These methods allow you to find the values of x that satisfy the equation and make it equal to zero.
 2. What is the quadratic formula?
Ans. The quadratic formula is a formula used to solve quadratic equations of the form ax^2 + bx + c = 0. It states that the solutions for x can be found using the formula: x = (-b ± √(b^2 - 4ac)) / (2a). This formula allows you to find the roots of any quadratic equation.
 3. Can all quadratic equations be factored?
Ans. No, not all quadratic equations can be factored. Some quadratic equations have roots that are irrational or complex numbers, which cannot be factored using integers or rational numbers. In such cases, factoring may not be possible and other methods like the quadratic formula or completing the square can be used to find the solutions.
 4. How do I know if a quadratic equation has real solutions?
Ans. A quadratic equation has real solutions if the discriminant (b^2 - 4ac) is greater than or equal to zero. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution (a repeated root). If the discriminant is negative, the equation has no real solutions but may have complex solutions.
 5. Can I solve quadratic equations using graphs?
Ans. Yes, quadratic equations can be solved using graphs. By plotting the quadratic equation on a graph, you can find the x-intercepts, which represent the solutions to the equation. The x-intercepts are the points where the graph intersects the x-axis. By analyzing the graph, you can determine the number and nature of the solutions to the quadratic equation.

## Mathematics (Maths) Class 10

120 videos|463 docs|105 tests

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