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Form a quadratic poly nomial One of whose zero is
2+sqrt3 & sum of zeroes is 4 .

Form a quadratic poly nomial One of whose zero is $2+\sqrt{3}$ $\&$ sum of zeroes is $4$ .

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Write a quadratic function from its zeros

Full solution

Q. Form a quadratic poly nomial One of whose zero is

2+\sqrt{3}

\&

sum of zeroes is

4

.

Identify Zero and Sum: Identify the given zero and the sum of the zeroes. $\newline$ We are given one zero of the quadratic polynomial as $2+\sqrt{3}$ and the sum of the zeroes as $4$ .
Find Other Zero: Find the other zero using the sum of the zeroes. $\newline$ If the sum of the zeroes is $4$ and one of the zeroes is $2+\sqrt{3}$ , then the other zero, let's call it 'b', can be found using the equation: $\newline$ Sum of zeroes = Zero $1$ + Zero $2$ $\newline$ $4 = (2+\sqrt{3}) + b$ $\newline$ Now, solve for 'b': $\newline$ $b = 4 - (2+\sqrt{3})$ $\newline$ $b = 2 - \sqrt{3}$
Write Quadratic Polynomial: Write the quadratic polynomial using the zeroes. $\newline$ The quadratic polynomial can be written in its factored form as: $\newline$ $f(x) = (x - \text{Zero 1})(x - \text{Zero 2})$ $\newline$ Substitute the zeroes into the equation: $\newline$ $f(x) = (x - (2+\sqrt{3}))(x - (2-\sqrt{3}))$
Expand Factored Form: Expand the factored form to get the standard form of the quadratic polynomial. $\newline$ $f(x) = (x - 2 - \sqrt{3})(x - 2 + \sqrt{3})$ $\newline$ Now, use the difference of squares formula, which states that $(a - b)(a + b) = a^2 - b^2$ : $\newline$ $f(x) = (x - 2)^2 - (\sqrt{3})^2$ $\newline$ $f(x) = (x^2 - 4x + 4) - 3$ $\newline$ $f(x) = x^2 - 4x + 4 - 3$ $\newline$ $f(x) = x^2 - 4x + 1$

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