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Algebra

Quadratic Equations

ALL MODULES

CAT 2025 Lesson : Quadratic Equations - Other Types

4.6 Other Types

Example 20

a, b, c

and

d

are positive integers that are in increasing Arithmetic Progression. If

a

and

d

are the roots of

x^{2} + mx + 10 = 0

, while

b

and

c

are the roots of

x^{2} + nx + 28 = 0

, then

m + n =

Solution

Product of roots are known in both the equations.
⇒

ad = \bm{10}

and

bc = \bm{28}

a, b, c

and

d

are positive integers in AP, we start by writing

ad

as product of integers.

ad = 1 \times 10 = 2 \times 5

a = 1

and

d = 10

, then the other

2

terms in AP are

b = 4

and

c = 7

. Here,

bc = 28

(Accepted)

If

a = 2

and

d = 5

, then the other

2

terms in AP are

b = 3

and

c = 4

. Here,

bc = 12

(Rejected)

∴

a = 1, b = 4, c = 7

and

d = 10

are the

4

roots.

m = -(a + d) = -11

and

n = -(b + c) = -11

∴

m + n = -22

Answer:

-22

Example 21

3^{x} = 9^{(\frac{y}{4})} = 27^{(\frac{2z}{3})}

and

xyz = 64

, then

x + 2y + 7 =

Solution

3^{x}

= 9^{(\frac{y}{4})}

= 27^{(\frac{2z}{3})}

⇒

3^{x} = 3^{(2 \times \frac{y}{4})}

⇒

3^{(3 \times \frac{2z}{3})}

⇒

3^{x} = 3^{\frac{y}{2}} = 3^{2z}

⇒

x = \dfrac{y}{2} = 2z

⇒

y = 2x

and

z = \dfrac{x}{2}

xyz = 64

⇒

x \times 2 x \times \dfrac{x}{2} = 64

⇒

x = 4

and

y = 8

x + 2y + 7 = 4 + 2 \times 8 + 7 = 27

Answer:

27

Example 22

Some kids were playing with marbles, and they realised that the average number of marbles with each kid was

4

more than the number of kids. If the total marbles was

320

, then what is the number of kids?

Solution

Let the number of kids be

x

.
∴ Average number of marbles

= x + 4

x(x + 4) = 320

⇒

x^{2} + 4x - 320 = 0

⇒

(x + 20) (x - 16) = 0

⇒

x = 16, -20

x

cannot be negative, number of kids

= \bm{16}

Alternatively

As the average and number of kids are integers, we need to express

320

as a product of

2

numbers, such that their difference is

4

320 = 20 \times 16

As the average is

4

more than the number of students,

20

is the average and 16 is the number of students.

Answer:

16

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