Question

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Answer 1

Answer:

Step-by-step explanation:

Alpha=a=2+√3

Beta=B=2-√3

So a+B=2,a*b=1

So equation:

x²-x(a+B)+a*b=0

So equation is x²-2x+1=0

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Answer 2

Answer:

x² - 4x + 1 = 0

(x-1)²

Step-by-step explanation:

Given,

• roots: 2+√3 and 2-√3

• Write quadratic equation

Solution1

• (x-(2+√3))(x-(2-√3))=

• x²- (2-√3)x - (2+√3)x + (2+√3)(2-√3)=

• x² - 4x + 1 = 0

Solution2

Sum of roots and product of roots in ax²+bx+c=0 are:

• -b/a and c/a

Sum of roots:

• 2+√3 + 2-√3 = 4

Product of roots:

• (2+√3) * (2-√3)= 4-3=1

• -b/a=4 => b= -4a

• c/a=1 => c=a

• Accepting c=a=1, we get b= -4

Then the equation is:

Anwer: x² - 4x + 1=0

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Divide x³-4x²+5x-2 by x-2

• x³-4x²+5x-2=
• x³-2x²- 2x²+4x+x-2=
• x²(x-2)-2x(x-2)+(x-2)=
• (x-2)(x²-2x+1)=
• (x-2)(x-1)²

(x³-4x²+5x-2) ÷ (x-2) = (x-1)²

Answer: (x-1)²