Question

Find the difference of roots of equation
x? – √20x + 4 = 0
(1) 4
(2) 2
(3) 3
(4) 5​

Answer 1

Answer:

$\huge \mathfrak \colorbox {lime}{answer}$

Well, although others have already posted the correct answer to this question, yet I'm gonna answer it anyway because I have a different and easier (according to me) method, than the ones mentioned. Here we go!

Given polynomial p(x) = x²-px+8

Let the zeroes be α and β respectively.

We know that α + β = -(coefficient of x)÷coefficient of x²

Therefore, α + β = - (-p)÷1

=> α + β = p …(I)

Also, it's given that, α - β = 4 …(II)

Also, αβ = constant term÷coefficient of x²

Therefore, αβ = 8÷1 = 8 …(III)

Now, squaring both sides in (I):

=> (α + β)² = p²

=> α² + β² + 2αβ = p²

=> (α - β)² + 2αβ + 2αβ = p² [∵α² + β² = (α-β)² + 2αβ]

=> (α - β)² + 4αβ = p²

=> (4)² + (4 × 8) = p² [By (II) and (III)]

=> 16 + 32 = p²

=> p = ±√48

=> p = 4√3 or -4√3

Answer 2

Therefore correct answer is 2 i.e, option 2 is the correct answer

Step-by-step explanation:

Find the difference of roots of equation

${x}^{2} - \sqrt{20} x + 4 = 0$

therefore

√5±1 are the roots of the quadratic equation

${x}^{2} - \sqrt{20} x + 4 = 0$

therefore x = √5+1 ,√5-1

it means

$\alpha = \sqrt{5} + 1 \\ \beta = \sqrt{5} - 1$

difference between their roots =

$\alpha - \beta = ( \sqrt{5} + 1) - ( \sqrt{5} - 1)$

$\alpha - \beta = \sqrt{5} + 1 - \sqrt{5} + 1 \\ \alpha - \beta = 1 + 1 = 2 \\ \alpha - \beta = 2$

therefore difference of the roots of the equation

${x}^{2} - \sqrt{20} x + 4 = 0$

is 2