Question

$\underline{\Huge {\purple{\sf Question}}}$

$\sf If\:roots\:of\:the\:equation\:2x^2-4x+2=0\:are\;\alpha\:and\:\beta\;,then\:prove\:that, \\ \sf \dfrac {\alpha}{\beta}+ \dfrac {\beta}{\alpha}+4\left(\dfrac {1}{\alpha} + \dfrac{1}{\beta}\right)+ 2\alpha\beta=12.$

$\underline{\sf Note:-}$

Answer only if you know.
$\maltese$Spammed answers will be deleted .​

Answer 1

Understanding the question:

Here we have given a quadratic equation and roots are ɑ and þ.So we can easily find the value of ɑ and þ by factorising the equation and after putting the values of ɑ and þ in LHS of proving equation,the required equation will be proved.

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So let's start! :D

➛2x²-4x+2=0

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➛2x²-2x-2x+2=0

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➛2x(x-1)-2(x-2)=0

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➛(x-1)(2x-2)=0

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x-1=0 ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ 2x-2=0

x=1 ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎2x=2

x=1 ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ x=1

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So both ‎ɑ and þ are 1 and 1.

We have to prove:-

✰ ɑ/þ+þ/ɑ+4( 1/ɑ+1/þ)+2ɑþ=12

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Put the value of ɑ and þ.

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➛1/1+1/1+ 4[(1)+(1)]+2(1)(1)=12

➛1+1+4(2)+2=12

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➛2+8+2=12

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➛12=12 ‎ ‎ ‎ ‎ ‎

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∴LHS=RHS

Answer 2

Required Answer:-

Given:

• The roots of the quadratic equation 2x² - 4x + 2 = 0 are α and β.

To Prove:

• α/β + β/α + 4(1/α + 1/β) + 2αβ = 12

Solution:

The general form of a quadratic equation is,

➡ ax² + bx + c = 0

In a quadratic equation, there can have a maximum of two roots. We are provided with a Quadratic equation whose roots are α and β.

Given equation,

➡ 2x² - 4x + 2 = 0

Taking 2 as common,

➡ 2(x² - 2x + 1) = 0

Dividing both sides by 2, we get,

➡ x² - 2x + 1 = 0

This is in the standard form. Now, roots of a quadratic equation are very nicely related to the coefficient of each term. Their relationship is given by,

➡ α + β = -b/a

➡ αβ = c/a

Here,

• a = coefficient of x²
• b = coefficient of x.
• c = coefficient of x⁰

In this equation,

• a = 1
• b = -2
• c = 1

Therefore,

➡ α + β = -b/a

➡ α + β = -(-2)/1

➡ α + β = 2

Again,

➡ αβ = c/a

➡ αβ = 1/1

➡ αβ = 1

Now, let's prove this.

Taking LHS,

α/β + β/α + 4(1/α + 1/β) + 2αβ

= (α² + β²)/αβ + 4[(α + β)/(αβ)] + 2αβ

= [(α + β)² - 2αβ]/αβ + 4[(α + β)/(αβ)] + 2αβ

Now, we know all the values. Substitute that,

= [(2² - 2 × 1)/1 + 4 × 2/1 + 2 × 1

= (4 - 2) + 8 + 2

= 2 + 8 + 2

= 12

= RHS (Hence Proved)