Answer:
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Answer:
x = (2 + √3) , (2 - √3)
Step-by-step explanation:
<u>GIVEN :-</u>
- A quadratic polynomial x² - 4x + 1
<u>TO FIND :-</u>
- Roots of the quadratic polynomial
<u>GENERAL FORMULAE TO BE USED IN THIS QUESTION :-</u>
<em>Quadratic formulae -</em>
For a polynomial ax² + bx + c , its roots are :-
![x = \frac{-b + \sqrt{b^2 - 4ac} }{2a} \; ; \frac{-b - \sqrt{b^2 - 4ac} }{2a}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B-b%20%2B%20%5Csqrt%7Bb%5E2%20-%204ac%7D%20%7D%7B2a%7D%20%5C%3B%20%3B%20%5Cfrac%7B-b%20-%20%5Csqrt%7Bb%5E2%20-%204ac%7D%20%7D%7B2a%7D)
<u>SOLUTION :-</u>
Use the quadratic formulae to find the roots of the polynomial.
![=> x = \frac{-(-4) + \sqrt{(-4)^2 - 4 \times 1 \times 1c} }{2 \times 1} \; ; \frac{-(-4) - \sqrt{(-4)^2 - 4 \times 1 \times 1} }{2 \times 1}](https://tex.z-dn.net/?f=%3D%3E%20x%20%3D%20%5Cfrac%7B-%28-4%29%20%2B%20%5Csqrt%7B%28-4%29%5E2%20-%204%20%5Ctimes%201%20%5Ctimes%201c%7D%20%7D%7B2%20%5Ctimes%201%7D%20%5C%3B%20%3B%20%5Cfrac%7B-%28-4%29%20-%20%5Csqrt%7B%28-4%29%5E2%20-%204%20%5Ctimes%201%20%5Ctimes%201%7D%20%7D%7B2%20%5Ctimes%201%7D)
![= \frac{4 + \sqrt{16 - 4} }{2} \; ; \frac{4- \sqrt{16 - 4}}{2}](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B4%20%2B%20%5Csqrt%7B16%20-%204%7D%20%7D%7B2%7D%20%5C%3B%20%3B%20%5Cfrac%7B4-%20%5Csqrt%7B16%20-%204%7D%7D%7B2%7D)
![= \frac{4 + \sqrt{12} }{2} \; ; \frac{4- \sqrt{12}}{2}](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B4%20%2B%20%5Csqrt%7B12%7D%20%7D%7B2%7D%20%5C%3B%20%3B%20%5Cfrac%7B4-%20%5Csqrt%7B12%7D%7D%7B2%7D)
![= \frac{4 + 2\sqrt{3} }{2} \; ; \frac{4- 2\sqrt{3}}{2}](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B4%20%2B%202%5Csqrt%7B3%7D%20%7D%7B2%7D%20%5C%3B%20%3B%20%5Cfrac%7B4-%202%5Csqrt%7B3%7D%7D%7B2%7D)
![= \frac{2(2 + \sqrt{3})}{2} \; ; \frac{2(2- \sqrt{3})}{2}](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B2%282%20%2B%20%5Csqrt%7B3%7D%29%7D%7B2%7D%20%5C%3B%20%3B%20%5Cfrac%7B2%282-%20%5Csqrt%7B3%7D%29%7D%7B2%7D)
![= (2 + \sqrt{3} ) \; ; (2 - \sqrt{3} )](https://tex.z-dn.net/?f=%3D%20%282%20%2B%20%5Csqrt%7B3%7D%20%29%20%5C%3B%20%3B%20%282%20-%20%5Csqrt%7B3%7D%20%29)