Question

if (3x - 5y) = 10 and xy = 5, find the value of
${9x}^{2} + {25y}^{2}$
​

Answer 1

Answer:

Q22 If 3x+5y = 11 and xy = 2, find the value of 9x^2 +25y^2.

Given that,3x + 5y = 11Squaring both sides, we get(3x + 5y)2 = (11)2(3x)2 + (5y)2 + 2 (3x) (5y) = 1219x2 + 25y2 + 30xy = 1219x2 + 25y2 + 30 (2) = 1219x2 ... More

Answer 2

Given -

• $\tt{3x + 5y = 11}$
• $\tt{xy = 5}$

To find -

• the value of $\tt{{9x}^{2} + {25y}^{2}}$

Solution -

Squaring both sides, we get

$\:\:\:\:\implies\sf{(3x + 5y) ^ 2 = (11) ^ 2}$

$\:\:\:\:\implies\sf{(3x) ^2 + (5y) ^2 + 2(3x) \times (5y) = 121}$

$\:\:\:\:\implies\sf{9x ^ 2 + 25y ^ 2 + 30xy = 121}$

$\:\:\:\:\implies\sf{9x ^ 2 + 25y ^ 2 + 30(2) = 121}$

$\:\:\:\:\implies\sf{9x ^ 2 + 25y ^ 2 = 121 - 60}$

$\:\:\:\:\implies\boxed{\bf{9x ^ 2 + 25y ^ 2 = 61}}$

Algebraic Identity used -

$\bigstar{\boxed{\sf{(a+b)^2=a^2+b^2+2ab}}}$

More Identities we should know -

$\bigstar\:\:\:$ (a + b)²=a² + 2ab + b²=( - a- b)²

$\bigstar\:\:\:$ (a - b)² = a² - 2ab + b²

$\bigstar\:\:\:$ (a - b)(a + b) = a² - b²

$\bigstar\:\:\:$ (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

$\bigstar\:\:\:$ (a + b - c)² = a² + b² + c² + 2ab - 2bc - 2ca

$\bigstar\:\:\:$ (a - b + c)² = a² + b² + c² - 2ab - 2bc + 2ca

$\bigstar\:\:\:$(- a + b + c)² = a² + b² + c²- 2ab + 2bc - 2ca

$\bigstar\:\:\:$ (a - b - c)² = a² + b² + c² - 2ab + 2bc - 2ca

$\bigstar\:\:\:$ (a + b)³ = a³ + b³ + 3ab(a + b)

$\bigstar\:\:\:$ (a - b)³ = a³ - b³- 3ab(a - b)

$\bigstar\:\:\:$ a³ + b³ = (a + b)³ - 3ab(a + b)= (a + b) (a²ab + b²)

$\bigstar\:\:\:$ a³ - b³ = (a - b)³ + 3ab (a - b) = (a − b) (a² + ab + b²)

$\bigstar\:\:\:$ a³ + b³ + c³-3abc = (a + b + c) (a² + b²+c²- ab-bc-ca) if a+b+c=0 then a³ + b3 + c³ = 3abc