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Solutions of Equations by Numerical Methods Change of sign Method Decimal Search Method f(x)=x³+2x²-3x+2 The solutions of f(x)=0 are the place where the curve cuts the x-axis. The graph shows that there is one real root lying in the interval [-4,-3] f(-4) = -64+32+12+2 = -18 f(-3 )= -27+18+9+2 = 2 The function f(x) is continuous therefore the change of sign form negative to positive in moving from x=-4 to x=-3 shows that there must be a root between the two values. The decimal search method will locate more accurate values of the root systematically by taking increments of size 0.1 in the interval [-4,-3] until a change of sign occurs. x -4 -3.9 -3.8 -3.7 -3.6 -3.5 -3.4 -3.3 -3.2 -3.1 -3 f(x) -18.000 -15.199 -12.592 -10.173 -7.936 -5.875 -3.984 -2.257 -0.688 0.729 2.000 There is a change of sign between -3.2 and -3.1, this show that the root is in the interval [-3.2,-3.1] Continuing the decimal search method will increase the accuracy by narrowing down the interval further. x -3.20 -3.19 -3.18 -3.17 -3.16 -3.15 -3.14 -3.13 -3.12 -3.11 -3.10 f(x) -0.688 -0.540 -0.393 -0.247 -0.103 0.039 0.180 0.320 0.457 0.594 0.729 x -3.16 -3.159 -3.158 -3.157 -3.156 -3.155 -3.154 -3.153 -3.152 -3.151 -3.15 f(x) -0.103 -0.089 -0.075 -0.060 -0.046 -0.032 -0.018 -0.003 0.011 0.025 0.039 x -3.153 -3.1529 -3.1528 -3.1527 -3.1526 -3.1525 -3.1524 -3.1523 -3.1522 -3.1521 -3.152 f(x) -0.003 -0.002 -0.001 0.001 0.002 0.004 0.005 0.007 0.008 0.009 0.011 x -3.1528 -3.15279 -3.15278 -3.15277 -3.15276 -3.15275 -3.15274 -3.15273 -3.15272 -3.15271 -3.1527 f(x) -0.00060 -0.00046 -0.00032 -0.00018 -0.00003 0.00011 0.00025 0.00039 0.00053 0.00068 0.00082 The decimal search method has shown that the root lies within the interval [-3.15276,-3.15275] The root...

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