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Find the solutions of the following equations: a) (i) x^(2)-3x+1=0

Find the solutions of the following equation:\newlinex2−3x+1=0 x^{2}-3 x+1=0

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Full solution

Q. Find the solutions of the following equation:\newlinex2−3x+1=0 x^{2}-3 x+1=0
  1. Identify coefficients: Identify coefficients aa, bb, and cc from the equation x2−3x+1=0x^2 - 3x + 1 = 0. \newlinea=1a = 1, b=−3b = -3 and c=1c = 1.
  2. Use quadratic formula: Use the quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \newlineSubstitute a=1a = 1, b=−3b = -3, c=1c = 1 into the formula. \newlinex=−(−3)±(−3)2−4⋅1⋅12⋅1x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4\cdot 1\cdot 1}}{2\cdot 1}
  3. Simplify expression: Simplify the expression under the square root. \newline9−4=5\sqrt{9 - 4} = \sqrt{5}
  4. Continue with formula: Continue with the quadratic formula.\newlinex=3±52x = \frac{3 \pm \sqrt{5}}{2}
  5. Write roots in simplest form: Write the roots in simplest form.\newlineRoots are x=3+52x = \frac{3 + \sqrt{5}}{2} and x=3−52x = \frac{3 - \sqrt{5}}{2}

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