Question

Given,
${x}^{2} + {y}^{2} = 29$
$xy = 2$
find
${x}^{4} + {y}^{4}$
​

Answer 1

Given Information:

$\rm\hookrightarrow x^{2}+y^{2}=29$

$\rm\hookrightarrow xy=2$

To Determine:

• The value of x⁴ + y⁴

Solution:

We have:

$\rm\longrightarrow x^{2}+y^{2}=29$

Squaring both sides, we get:

$\rm\longrightarrow (x^{2}+y^{2})^{2}=29^{2}$

Using identity (a + b)² = a² + 2ab + b², we get:

$\rm\longrightarrow x^{4}+y^{4}+2\cdot x^{2}\cdot y^{2}=841$

$\rm\longrightarrow x^{4}+y^{4}+2\cdot(xy)^{2}=841$

$\rm\longrightarrow x^{4}+y^{4}+2\cdot (2)^{2}=841$

$\rm\longrightarrow x^{4}+y^{4}+8=841$

$\rm\longrightarrow x^{4}+y^{4}=833$

Which is our required answer.

Learn More:

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• a² - b² = (a + b)(a - b)
• (a + b)³ = a³ + 3ab(a + b) + b³
• (a - b)³ = a³ - 3ab(a - b) - b³
• a³ + b³ = (a + b)(a² - ab + b²)
• a³ - b³ = (a - b)(a² + ab + b²)
• (x + a)(x + b) = x² + (a + b)x + ab
• (x + a)(x - b) = x² + (a - b)x - ab
• (x - a)(x + b) = x² - (a - b)x - ab
• (x - a)(x - b) = x² - (a + b)x + ab