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I am trying to write a function in C++ that solves for X using the quadratic equation. This is what I have written initially, which seems to work as long as there are no complex numbers for an answer:

float solution1 = (float)(-1.0 * b) + (sqrt((b * b) - (4 * a * c)));
solution1 = solution1 / (2*a);

cout << "Solution 1: " << solution1 << endl;

float solution2 = (float)(-b) - (sqrt((b*b) - (4 * a * c)));
solution2 = solution2 / (2*a);
cout << "Solution 2: " << solution2;

If, for example, I use the equation: x^2 - x - 6, I get the solution 3, -2 correctly.

My question is how would I account for complex numbers....for example, given the equation:

x^2 + 2x + 5

Solving by hand, I would get -1 + 2i, -1 - 2i.

Well, I guess two question, can I write the above better and also make it account for the complex number?

Thanks for any help!

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    The other guys provided good answers so no reason for me to try and outshine them ;) However, if you want a more general solution to the equation ax^2+bx+c=0, remember that a==0 should be a valid value. This would result in a division of zero, so you must take care of this case separately. In this case it would mean that you are left with a linear equation with one root. Cheers ! – ralphtheninja May 22 '09 at 14:38
  • You imply you are concerned about complex roots, but what about complex coefficients in the original equation? – Jouni K. Seppänen May 22 '09 at 14:47
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    Complex coefficients requires another approach completely. So that will be the next question :) *scurrs off and prepares an answer before hand* – ralphtheninja May 22 '09 at 16:20

6 Answers6

22

An important note to all of this. The solutions shown in these responses and in the original question are not robust.

The well known solution (-b +- sqrt(b^2 - 4ac)) / 2a is known to be non-robust in computation when ac is very small compered to b^2, because one is subtracting two very similar values. It is better to use the lesser known solution 2c / (-b -+ sqrt(b^2 -4ac)) for the other root.

A robust solution can be calculated as:

temp = -0.5 * (b + sign(b) * sqrt(b*b - 4*a*c);
x1 = temp / a;
x2 = c / temp;

The use of sign(b) ensures that we are not subtracting two similar values.

For the OP, modify this for complex numbers as shown by other posters.

Andrew Stein
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7

Something like this would work:

struct complex { double r,i; }
struct pair<T> { T p1, p2; }

pair<complex> GetResults(double a, double b, double c)
{
  pair<complex> result={0};

  if(a<0.000001)    // ==0
  {
    if(b>0.000001)  // !=0
      result.p1.r=result.p2.r=-c/b;
    else
      if(c>0.00001) throw exception("no solutions");
    return result;
  }

  double delta=b*b-4*a*c;
  if(delta>=0)
  {
    result.p1.r=(-b-sqrt(delta))/2/a;
    result.p2.r=(-b+sqrt(delta))/2/a;
  }
  else
  {
    result.p1.r=result.p2.r=-b/2/a;
    result.p1.i=sqrt(-delta)/2/a;
    result.p2.i=-sqrt(-delta)/2/a;
  }

  return result;
}

That way you get the results in a similar way for both real and complex results (the real results just have the imaginary part set to 0). Would look even prettier with boost!

edit: fixed for the delta thing and added a check for degenerate cases like a=0. Sleepless night ftl!

Blindy
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  • If the sqrt succeeds, the result is >= 0. And if the argument is negative, your program crashes. You should test the sign first, and calculate the sqrt afterwards. If the sign would be negative, you'd set result.first.i = +sqrt(4*a*c-b*b)/2/a. (Why define your own type if there's a perfectly fine std::pair ?) – MSalters May 22 '09 at 14:56
  • should be delta=b*b-4*a*c, and take sqrt only when delta>=0. delta=0 or a=0 are valid cases when we have one root. What if a=b=0 and c=1? – Eric Bainville May 22 '09 at 14:59
  • What if? That's not a quadratic function, and the /2/a part will fail. This works reasonably well if delta=0, except that tou'll return the same result twice. – MSalters May 22 '09 at 15:07
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    Why are you defining a pair and a complex type? Both already exist! – Harper Shelby May 22 '09 at 19:07
  • Portable? Idk.. I wrote it all in the text editor after like 24 hours of no sleep, I'm surprised it even compiles <.> – Blindy May 22 '09 at 23:53
  • I would have gone for an intermediate variable to store sqrt(delta): the engineer in me cries whenever he sees unnecessary duplication of a floating point operation... – Jaime May 23 '09 at 18:04
  • Sorry about the late comment but a new user is using this code and they are getting errors. Why are you using `sqrt(delta)` when `delta < 0`? Most implementations will return `NaN` in that case. – NathanOliver Sep 05 '17 at 15:53
  • Must have been a typo, it's supposed to be `sqrt(-delta)`, ie the square root of the absolute value of delta. – Blindy Sep 12 '17 at 17:59
4

You more or less have it, just check to see if the part that's inside the square root is negative and then keep track of that separately in your reductions.

Jeff Moser
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3

You could basically just use std::complex<float> instead of float to get support for complex numbers.

sth
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1

Nicking the idea from Blindy:

typedef std::complex<double> complex;
using std::pair;
pair<complex> GetResults(double a, double b, double c)
{
  double delta=(b*b-4*a*c);
  double inv_2a = 1/2/a;
  if(delta >= 0) {
    double root = sqrt(delta);
    return std::make_pair(
        complex((-b-root)*inv_2a),
        complex((-b+root)*inv_2a);
  } else {
    double root = sqrt(-delta);
    return std::make_pair(
        complex(-b*inv_2a, -root*inv_2a)),
        complex(-b*inv_2a, +root*inv_2a)));
  }
}
MSalters
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-1

I tried the program without using 'math.h' header and also tried different logic...but my program can answer only those quadratic equations which have coefficient of 'x square' as one ..... and where coefficient of 'x' can be expressed as an addition of two numbers which are factors of constant term. eg. x square +8x+16; x square +7x+12; etc. here 8=4+4 & 16=4*4; here coefficient of x can be expressed as an addition of two numbers which are factors of constant term 16... I myself is not fully satisfied with it but tried something different, without using the formula for solving quadratic equation. code is;

        #include<iostream.h>
        #include<conio.h>
         class quadratic
              {
                int b,c ;
                float l,k;
                public:
               void solution();
              };
        void quadratic::solution()
             {
                 cout<<"Enter coefficient of x and the constant term of the quadratic eqn where coefficient of x square is one";
                 cin>>b>>c;

                 for(l=1;l<b;l++)
                  {
                   for(k=1;k<b;k++)
                    {
                     if(l+k==b&&l*k==c)
                        {
                          cout<<"x="<<-l<<"\t"<<"or"<<"\t"<<"x="<<-k;
                          cout<<"\n";
                         }
                    }
                }
            }
              void main()
                 {
                  quadratic a;
                   clrscr();
                  a.solution();
                  getch();
                 }
Deepeshkumar
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