Question

If a2+b2 = 24and ab =10 find a(a+b)^2 +3(a-b)^2​

Answer 1

Answer:

The answer is 202

Step-by-step explanation:

$here \: 2 \: formula \: needs \: to \: solved \: \\ the \: problem$

$\\ 1. \: {a}^{2} + {b}^{2} = {(a + b)}^{2} - 2ab$

$\\ 1. \: {a}^{2} + {b}^{2} = {(a - b)}^{2} + 2ab$

I explained and solved the problem in picture.

Answer 2

Answer:

The value of the given expression is either 202.08 or 114.08.

Step-by-step-explanation:

We have given that,

• a² + b² = 24
• ab = 10

We have to find the value of

a ( a + b )² + 3 ( a - b )²

Let this expression be P.

∴ P = a ( a + b )² + 3 ( a - b )²

⇒ P = a ( a² + 2ab + b² ) + 3 ( a² - 2ab + b² )

⇒ P = a ( a² + b² + 2 * ab ) + 3 ( a² + b² - 2 * ab )

⇒ P = a ( 24 + 2 * 10 ) + 3 ( 24 - 2 * 10 )

⇒ P = a ( 24 + 20 ) + 3 ( 24 - 20 )

⇒ P = a * 44 + 3 * 4

⇒ P = 44a + 12

Now,

ab = 10

⇒ b = 10 / a

By using this value, we get,

a² + b² = 24

⇒ a² + ( 10 / a )² = 24

⇒ a² + ( 10² / a² ) = 24

⇒ a² + ( 100 / a² ) = 24

⇒ ( a² * a² + 100 ) / a² = 24

⇒ a⁴ + 100 = 24a²

⇒ a⁴ - 24a² + 100 = 0

By substituting a² = x, we get,

⇒ x² - 24x + 100 = 0

Comparing with ax² + bx + c = 0, we get,

• a = 1
• b = - 24
• c = 100

By quadratic formula,

$\displaystyle{\boxed{\pink{\sf\:x\:=\:\dfrac{-\:b\:\pm\:\sqrt{b^2\:-\:4ac}}{2a}\:}}}$

$\displaystyle{\implies{\sf\:x\:=\:\dfrac{-\:(\:-\:24\:)\:\pm\:\sqrt{(\:-\:24\:)^2\:-\:4\:\times\:1\:\times\:100}}{2\:\times\:1}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{24\:\pm\:\sqrt{576\:-\:400}}{2}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{24\:\pm\:\sqrt{176}}{2}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{24\:\pm\:13.266}{2}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{24\:+\:13.266}{2}\:\quad\:OR\:\quad\:x\:=\:\dfrac{24\:-\:13.266}{2}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{37.266}{2}\:\quad\:OR\:\quad\:x\:=\:\dfrac{10.734}{2}}$

$\displaystyle{\implies\sf\:x\:=\:18.633\:\quad\:OR\:\quad\:x\:=\:5.367}$

$\displaystyle{\therefore\:\boxed{\blue{\sf\:x\:\approx\:18.63\:}}\sf\quad\:OR\:\quad\:\boxed{\blue{\sf\:x\:\approx\:5.37\:}}}$

Now,

$\displaystyle{\sf\:a^2\:=\:x}$

$\displaystyle{\implies\sf\:a\:=\:\sqrt{x}}$

$\displaystyle{\implies\sf\:a\:=\:\sqrt{18.63}}$

$\displaystyle{\implies\sf\:a\:=\:4.316}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:a\:\approx\:4.32\:}}}}$

Also,

$\displaystyle{\sf\:a^2\:=\:x}$

$\displaystyle{\implies\sf\:a\:=\:\sqrt{x}}$

$\displaystyle{\implies\sf\:a\:=\:\sqrt{5.37}}$

$\displaystyle{\implies\sf\:a\:=\:2.317}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:a\:\approx\:2.32\:}}}}$

Now, by using a = 4.32,

P = 44a + 12

⇒ P = 44 * 4.32 + 12

⇒ P = 190.08 + 12

⇒ P = 202.08

Now, by using a = 2.32,

P = 44a + 12

⇒ P = 44 * 4.32 + 12

⇒ P = 102.08 + 12

⇒ P = 114.08

∴ The value of the given expression is either 202.08 or 114.08.