Solving equations

> q := 3*x^2 -5*x +2;

                                 2
                         q := 3 x  - 5 x + 2

> solve(q=0);

                                2/3, 1

1. Use subs to check your result:
> subs(x=1,q);

                                  0

Sometimes solutions are irrational numbers:
> solve(x^2 -3=0);

                              1/2    1/2
                             3   , -3

> solve(3*x^2 -5*x +1=0);

                               1/2              1/2
                   5/6 + 1/6 13   , 5/6 - 1/6 13

2. You may wish to have decimal approximation for these solutions:
> fsolve(3*x^2 -5*x +1=0);

                       .2324081208, 1.434258546

3. You acess the student package using the with statement:
> with(student);

[D, Diff, Doubleint, Int, Limit, Lineint, Product, Sum, Tripleint,

    changevar, combine, completesquare, distance, equate, extrema,

    integrand, intercept, intparts, isolate, leftbox, leftsum,

    makeproc, maximize, middlebox, middlesum, midpoint, minimize,

    powsubs, rightbox, rightsum, showtangent, simpson, slope,

    trapezoid, value]

> 3*x^2 -5*x +1=0;

                             2
                          3 x  - 5 x + 1 = 0

> completesquare(",x);

                                   2   13
                        3 (x - 5/6)  - -- = 0
                                       12

> isolate(",x-5/6);

                                          2
                    x - 5/6 = RootOf(36 _Z  - 13)

> isolate(",x);

                                    2
                    x = RootOf(36 _Z  - 13) + 5/6

3. Some equations have complex solutions:
> solve(x^2 +1=0);

                                I, -I

> solve(3*x^2 -5*x +7=0);

                               1/2                1/2
                 5/6 + 1/6 I 59   , 5/6 - 1/6 I 59

> fsolve(3*x^2 -5*x +7=0);
The fsolve command finds only real numbers solutions.

> solve(3*x^2 -5*x +7=0);

                               1/2                1/2
                 5/6 + 1/6 I 59   , 5/6 - 1/6 I 59

> s := ";

                                 1/2                1/2
              s := 5/6 + 1/6 I 59   , 5/6 - 1/6 I 59

4. The variable s is a sequence with two elements.
You can acess each solution separetely:
> eval(s[1]);

                                        1/2
                          5/6 + 1/6 I 59

> evalf(s[1]);

                     .8333333333 + 1.280190958 I

> eval(s[2]);

                                        1/2
                          5/6 - 1/6 I 59

> evalf(s[2]);

                     .8333333333 - 1.280190958 I

5. You can solve polynomial equations of higher degree than 2.:
> q := 6*x^4 -35*x^3 +22*x^2 +17*x -10;

                        4       3       2
                q := 6 x  - 35 x  + 22 x  + 17 x - 10

> solve(q=0);

                           1, -2/3, 5, 1/2

> solve(q=1);

                        4        3        2
             RootOf(6 _Z  - 35 _Z  + 22 _Z  + 17 _Z - 11)

> fsolve(q=1);

         -.6814339806, .5708552757, .9422797802, 5.001632258

> plot(q-1, x=-1..2);

Other Types of Equations

1. You can solve many trigonometric equations:
> p := cos(x) -sin(x);

                         p := cos(x) - sin(x)

> solve(p=0);

                                1/4 Pi

2. The solve command usually returns a single solution to 
nonpolynomial equations even if there are many solutions.

> plot(p,x=0..2*Pi);

> fsolve(p=0, x=1.5..4);

                             3.926990817

> fsolve(p=0,x, 1.5..4);

                             3.926990817

> evalf("/Pi);

                             1.250000000

3. You ca solve logarithmic equations also:
> solve(ln(x) + ln(x+1) = ln(2));

                                  1

> solve(2^x=5);

                                ln(5)
                                -----
                                ln(2)

> fsolve(2^x=5);

                             2.321928095

4. You can solve inequalities:
> solve(x^2 -5*x < 0);

                     RealRange(Open(0), Open(5))

> solve(x^2 -5*x >=0);

           RealRange(-infinity, 0), RealRange(5, infinity)

5. You can also solve systems of equations:
> s := solve ({x+y=5, x-y=2},{x,y});

                       s := {y = 3/2, x = 7/2}

> subs(",x+y=5);

                                5 = 5

> plot({5-x,x-2},x=-4..4);

> solve({x+y=5., x-y=2},{x,y});

                  {x = 3.500000000, y = 1.500000000}

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