Question

If A and B are the roots of x²-2x + 3 = 0,
then the equation with roots A +2,B+ 2 is​

Answer 1

Question

If A and B are the roots of x²-2x + 3 = 0,then the equation with roots A +2,B+ 2 is____

Solution

p(x) = x² - 6x + 11

Explanation :

A and B are the zero of the polynomial,p(x) = x² - 2x + 3 = 0

To finD :

Polynomial whose zeros are (A + 2) and (B + 2)

There are two approaches to solve the above Question

• We find the zeros of the first polynomial and thn add them accordingly and find the polynomial

• We find the sum and product of zeros and accordingly find the sum and product of the required polynomial

Note

Sum of Zeros : - x coefficient/x² coefficient

Product of Zeros : constant term /x² coefficient

Here,

Sum of Zeros

A + B = -(-2)/1

→ A + B = 2

Product of Zeros

→ AB = 3

Let S and P denote the Sum and Product of Zeros of Required Polynomial

Now,

Sum of Zeros

S = (A + 2) + (B + 2)

» S = (A + B) + 4

» S = 6

Product of Zeros

P = (A + 2)(B + 2)

» P = 2² + 2(A + B) + AB

» P = 4 + 2(2) + 3

» P = 11

Required Polynomial

p(x) = x² - Sx + P

→ p(x) = x² - 6x + 11

Answer 2

$\huge {\red{\boxed{ \overline{ \underline{ \mid\mathfrak{An}{\mathrm{sw}{ \sf{er}} \colon\mid}}}}}}$

Given :

• Quadratic Equation is x² - 2x + 3 = 0
• Roots are A and B

Solution :

As we know that ax² + bx + c is general form of a quadratic equation.

So, compare given equation with it

We get,

a = 1

b = -2

c = 3

_________________________________

Now use formula for sum of zeroes :

$\large \star {\boxed{\sf{Sum \: of \: zeroes \: = \: \dfrac{-b}{a}}}} \\ \\ \implies {\sf{A \: + \: B \: = \: \dfrac{-(-2)}{1}}} \\ \\ \implies {\sf{A \: + \: B \: = \: 2}}$

Any now use formula for Product of Roots :

$\large \star {\boxed{\sf{Product \: = \: \dfrac{c}{a}}}} \\ \\ \implies {\sf{AB \: = \: \dfrac{3}{1}}} \\ \\ \implies {\sf{AB \: = \: 3}}$

$\rule{200}{2}$

Now, We have to find a quadratic equation whose zeroes are A + 2 , B + 2

For that take sum and Product

Sum of Zeroes

» A + 2 + B + 2

» A + B + 4

» 2 + 4

» 6 (Sum of zeroes is 6)

________________________

Product Of Zeroes

» (A + 2)(B + 2)

» AB + 2B + 2A + 4

» AB + 4 + 2(A + B)

» 3 + 4 + 2(2)

» 3 + 4 + 4

» 11 (Product of zeroes is )

$\rule{100}{2}$

We have formula for equation of quadratic equation :

$\large \star {\boxed{\sf{x^2 \: - \: (Sum \: of \: Roots)x \: + \: Product}}} \\ \\ \implies {\sf{x^2 \: - \: (6)x \: + \: 11}} \\ \\ \implies {\sf{x^2 \: - \: 6x \: + \: 11}}$

Equation is x² - 6x + 11