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jawab menggunakan metode regula falsi: −0,6x2+2.4x+5,5=0-0,6x^2 + 2.4x + 5,5 = 0 pada interval [5;10][ 5 ; 10 ]

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Q. jawab menggunakan metode regula falsi: −0,6x2+2.4x+5,5=0-0,6x^2 + 2.4x + 5,5 = 0 pada interval [5;10][ 5 ; 10 ]
  1. Define Function: Define the function f(x)=−0.6x2+2.4x+5.5f(x) = -0.6x^2 + 2.4x + 5.5.
  2. Calculate f(5)f(5) and f(10)f(10): Calculate f(5)f(5) and f(10)f(10).
    f(5)=−0.6(5)2+2.4(5)+5.5=−0.6(25)+12+5.5=−15+12+5.5=2.5f(5) = -0.6(5)^2 + 2.4(5) + 5.5 = -0.6(25) + 12 + 5.5 = -15 + 12 + 5.5 = 2.5
    f(10)=−0.6(10)2+2.4(10)+5.5=−0.6(100)+24+5.5=−60+24+5.5=−30.5f(10) = -0.6(10)^2 + 2.4(10) + 5.5 = -0.6(100) + 24 + 5.5 = -60 + 24 + 5.5 = -30.5
  3. Determine Root Interval: Since f(5)f(5) and f(10)f(10) have opposite signs, there is a root between 55 and 1010.
  4. Find Next Approximation x1x_1: Apply the regula falsi formula to find the next approximation x1x_1.
    x1=5−f(5)×(10−5)/(f(10)−f(5))x_1 = 5 - f(5) \times (10 - 5) / (f(10) - f(5))
    x1=5−2.5×(10−5)/(−30.5−2.5)x_1 = 5 - 2.5 \times (10 - 5) / (-30.5 - 2.5)
    x1=5−2.5×5/(−33)x_1 = 5 - 2.5 \times 5 / (-33)
    x1=5−12.5/(−33)x_1 = 5 - 12.5 / (-33)
    x1=5+12.5/33x_1 = 5 + 12.5 / 33
    x1=5+0.37878787878x_1 = 5 + 0.37878787878
    x1=5.37878787878x_1 = 5.37878787878
  5. Calculate f(x1)f(x_1): Calculate f(x1)f(x_1).
    f(5.37878787878)=−0.6(5.37878787878)2+2.4(5.37878787878)+5.5f(5.37878787878) = -0.6(5.37878787878)^2 + 2.4(5.37878787878) + 5.5
    f(5.37878787878)=−0.6(28.931)+12.909+5.5f(5.37878787878) = -0.6(28.931) + 12.909 + 5.5
    f(5.37878787878)=−17.3586+12.909+5.5f(5.37878787878) = -17.3586 + 12.909 + 5.5
    f(5.37878787878)=−17.3586+18.409f(5.37878787878) = -17.3586 + 18.409
    f(5.37878787878)=1.0504f(5.37878787878) = 1.0504
  6. Check Root Interval: Since f(5)f(5) and f(x1)f(x_1) have opposite signs, the root lies between 55 and x1x_1.
  7. Find Next Approximation x2x_2: Apply the regula falsi formula again to find the next approximation x2x_2.
    x2=5−f(5)×(x1−5)/(f(x1)−f(5))x_2 = 5 - f(5) \times (x_1 - 5) / (f(x_1) - f(5))
    x2=5−2.5×(5.37878787878−5)/(1.0504−2.5)x_2 = 5 - 2.5 \times (5.37878787878 - 5) / (1.0504 - 2.5)
    x2=5−2.5×0.37878787878/(−1.4496)x_2 = 5 - 2.5 \times 0.37878787878 / (-1.4496)
    x2=5−0.94696969695/(−1.4496)x_2 = 5 - 0.94696969695 / (-1.4496)
    x2=5+0.94696969695/1.4496x_2 = 5 + 0.94696969695 / 1.4496
    x2=5+0.6531986532x_2 = 5 + 0.6531986532
    x2=5.6531986532x_2 = 5.6531986532
  8. Find Next Approximation x2x_2: Apply the regula falsi formula again to find the next approximation x2x_2.
    x2=5−f(5)⋅(x1−5)/(f(x1)−f(5))x_2 = 5 - f(5) \cdot (x_1 - 5) / (f(x_1) - f(5))
    x2=5−2.5⋅(5.37878787878−5)/(1.0504−2.5)x_2 = 5 - 2.5 \cdot (5.37878787878 - 5) / (1.0504 - 2.5)
    x2=5−2.5⋅0.37878787878/(−1.4496)x_2 = 5 - 2.5 \cdot 0.37878787878 / (-1.4496)
    x2=5−0.94696969695/(−1.4496)x_2 = 5 - 0.94696969695 / (-1.4496)
    x2=5+0.94696969695/1.4496x_2 = 5 + 0.94696969695 / 1.4496
    x2=5+0.6531986532x_2 = 5 + 0.6531986532
    x2=5.6531986532x_2 = 5.6531986532Check if f(x2)f(x_2) is close enough to zero or if another iteration is needed.

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