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Factor the quadratic expression x^(2)+x+1=0

Factor the quadratic expression x2+x+1=0 x^{2}+x+1=0

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Full solution

Q. Factor the quadratic expression x2+x+1=0 x^{2}+x+1=0
  1. Identify Equation Type: Identify the type of equation we are dealing with.\newlineThe given equation x2+x+1=0x^{2} + x + 1 = 0 is a quadratic equation, which can be solved using the quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where aa, bb, and cc are the coefficients of the terms x2x^2, xx, and the constant term, respectively.
  2. Apply Quadratic Formula: Apply the quadratic formula to find the solutions for xx. For the given equation, a=1a = 1, b=1b = 1, and c=1c = 1. Plugging these values into the quadratic formula gives us: x=−1±12−4⋅1⋅12⋅1x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} x=−1±1−42x = \frac{-1 \pm \sqrt{1 - 4}}{2} x=−1±−32x = \frac{-1 \pm \sqrt{-3}}{2}
  3. Simplify Square Root: Simplify the expression under the square root.\newlineSince the expression under the square root is negative, we have a complex number. The square root of −3-3 can be written as 3i\sqrt{3}i, where ii is the imaginary unit.\newlinex=−1±3i2x = \frac{-1 \pm \sqrt{3}i}{2}
  4. Write Final Solutions: Write the final solutions.\newlineThe equation has two complex solutions:\newlinex=−1+3i2x = \frac{-1 + \sqrt{3}i}{2} and x=−1−3i2x = \frac{-1 - \sqrt{3}i}{2}

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