Question

If 4x2+y2=40 and xy=6, find the value of 2x+y​

Answer 1

Answer:

The value of 2x + y is 8.

Step-by-step-explanation:

We have given that,

4x² + y² = 40

xy = 6

We have to find the value of 2x + y.

Now, we know that,

( x + y )² = x² + 2xy + y² - - [ Algebraic identity ]

∴ ( 2x + y )² = ( 2x )² + 2 * 2x * y + y²

⇒ ( 2x + y )² = 4x² + 4xy + y²

⇒ ( 2x + y )² = ( 4x² + y² ) + 4 * ( xy )

⇒ ( 2x + y )² = ( 40 ) + 4 * ( 6 ) - - [ Given ]

⇒ ( 2x + y )² = 40 + 24

⇒ ( 2x + y )² = 64

⇒ √[ ( 2x + y )² ] = √64 - - [ Taking square roots ]

⇒ ( 2x + y ) = 8

∴ 2x + y = 8

─────────────────────

Additional Information:

Some Algebraic Identities:

1. ( a + b )² = a² + 2ab + b²

2. ( a - b )² = a² - 2ab + b²

3. ( a + b ) ( a - b ) = a² - b²

4. ( a + b )³ = a³ + 3a²b + 3ab² + b³

5. a³ + b³ = ( a + b )³ - 3ab ( a + b )

6. a³ - b³ = ( a - b )³ + 3ab ( a - b )

7. ( a + b + c )² = a² + b² + c² + 2ab + 2bc + 2ac

Answer 2

♣ Qᴜᴇꜱᴛɪᴏɴ :

$\text { If } 4 \mathrm{x}^{2}+\mathrm{y}^{2}=40 \text { and } \mathrm{xy}=6, \text { find the value of } 2 \mathrm{x}+\mathrm{y}$

★═════════════════★

♣ ᴀɴꜱᴡᴇʀ :

$\large\boxed{\sf{2x+y=8}}$

★═════════════════★

♣ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴꜱ :

Use algebraic identity : (a + b)² = a² + b² + 2ab

$(2 \mathrm{x}+\mathrm{y})^{2}=(2 \mathrm{x})^{2}+\mathrm{y}^{2}+2 \times 2 \mathrm{x} \times \mathrm{y}$

$\Rightarrow(2 \mathrm{x}+\mathrm{y})^{2}=\left(4 \mathrm{x}^{2}+\mathrm{y}^{2}\right)+4 \mathrm{xy}$

We have value of xy = 6

$\Rightarrow(2 \mathrm{x}+\mathrm{y})^{2}=\left(4 \mathrm{x}^{2}+\mathrm{y}^{2}\right)+4 \times \mathrm{6}$

We have value of 4x² + y²  = 40

$\Rightarrow(2 \mathrm{x}+\mathrm{y})^{2}=40+24$

$\Rightarrow(2 \mathrm{x}+\mathrm{y})^{2}=64$

$\rm{\sqrt{2x+y}=\sqrt{64}}$

$\large\boxed{\sf{2x+y=8}}$