Quadratic Equation Analysis

Quadratic Equation Elements

  • Given the equation:
    4x2−6x−1=04x^2 - 6x - 1 = 0

Identification of Coefficients

  • Coefficient of x2x^2 (leading coefficient):
      - a=4a = 4
  • Coefficient of xx:
      - b=−6b = -6
  • Constant term:
      - c=−1c = -1

Application of the Quadratic Formula

  • The quadratic formula is used to find the roots of the equation and is given as:
    x=−bext±ext√(b2−4ac)2ax = \frac{-b ext{ ± } ext{√}(b^2 - 4ac)}{2a}

Calculation of the Discriminant

  • The discriminant, DD, is calculated as follows:
      - D=b2−4acD = b^2 - 4ac
      - Substituting values, we get:
      - D=(−6)2−4(4)(−1)D = (-6)^2 - 4(4)(-1)
      - Calculating further,
        - D=36+16=52D = 36 + 16 = 52

Roots of the Equation

  • Finding the roots using the quadratic formula:
      - x=−(−6)ext±ext√(52)2(4)x = \frac{-(-6) ext{ ± } ext{√}(52)}{2(4)}
      - Simplifying further:
      - x=6ext±ext√(52)8x = \frac{6 ext{ ± } ext{√}(52)}{8}
      - Recognizing that ext√(52)ext{√}(52) can be simplified:
      - ext√(52)=ext√(4imes13)=2ext√(13)ext{√}(52) = ext{√}(4 imes 13) = 2 ext{√}(13)

Final Expression of the Roots

  • Thus, the final roots can be expressed as:
      - x=6ext±2ext√(13)8x = \frac{6 ext{ ± } 2 ext{√}(13)}{8}
      - Which further reduces to:
      - x=3ext±ext√(13)4x = \frac{3 ext{ ± } ext{√}(13)}{4}

Conclusion

  • The roots of the equation 4x2−6x−1=04x^2 - 6x - 1 = 0 are:
      - x1=3+ext√(13)4x_1 = \frac{3 + ext{√}(13)}{4}
      - x2=3−ext√(13)4x_2 = \frac{3 - ext{√}(13)}{4}