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 Procedure Abstract Required_Reading Keywords Brief_I/O Detailed_Input Detailed_Output Parameters Exceptions Files Particulars Examples Restrictions Literature_References Author_and_Institution Version Index_Entries

Procedure

```   rquad_c ( Roots of a quadratic equation )

SpiceDouble  b,
SpiceDouble  c,
SpiceDouble  root1[2],
SpiceDouble  root2[2] )

```

Abstract

```   Find the roots of a quadratic equation.
```

```   None.
```

Keywords

```   MATH
POLYNOMIAL
ROOT

```

Brief_I/O

```   VARIABLE  I/O  DESCRIPTION
--------  ---  --------------------------------------------------
a          I   Coefficient of quadratic term.
b          I   Coefficient of linear term.
c          I   Constant.
root1      O   Root built from positive discriminant term.
root2      O   Root built from negative discriminant term.
```

Detailed_Input

```   a,
b,
c           are the coefficients of a quadratic polynomial

2
ax   +   bx   +   c.
```

Detailed_Output

```   root1,
root2       are the roots of the equation,

2
ax   +   bx   +   c   =  0.

root1 and root2 are both arrays of length 2. The
first element of each array is the real part of a
root; the second element contains the complex part
of the same root.

When a is non-zero, root1 represents the root

_____________
/  2
- b   +    \/  b    -   4ac
---------------------------
2a

and root2 represents the root

_____________
/  2
- b   -    \/  b    -   4ac
--------------------------- .
2a

When a is zero and b is non-zero, root1 and root2
both represent the root

- c / b.
```

Parameters

```   None.
```

Exceptions

```   1)  If the input coefficients `a' and `b' are both zero, the error
SPICE(DEGENERATECASE) is signaled. The output arguments are not
modified.
```

Files

```   None.
```

Particulars

```   None.
```

Examples

```   The numerical results shown for these examples may differ across
platforms. The results depend on the SPICE kernels used as
input, the compiler and supporting libraries, and the machine
specific arithmetic implementation.

1) Humor us and suppose we want to compute the "golden ratio."

The quantity `r' is defined by the equation

1/r = r/(1-r),

which is equivalent to

2
r   +  r  -  1  =  0.

The following code example does the job.

Example code begins here.

/.
./
#include <stdio.h>
#include "SpiceUsr.h"

int main( )
{
/.
Local variables.
./
SpiceDouble             root1 [ 2 ];
SpiceDouble             root2 [ 2 ];

/.
Compute "golden ratio."  The root we want,

___
/
-1 + \/  5
-----------,
2

is contained in root1.
./
rquad_c ( 1., 1., -1., root1, root2 );

/.
Print the results.
./
printf ( "The \"golden ratio\" is %f\n", root1[0] );

return ( 0 );

}

When this program was executed on a Mac/Intel/cc/64-bit
platform, the output was:

The "golden ratio" is 0.618034

2) Calculate the roots of the following quadratic equation:

2
x  + 2x + 3 = 0

Example code begins here.

/.
./
#include <stdio.h>
#include "SpiceUsr.h"

int main( )
{
/.
Local variables.
./
SpiceDouble             root1 [ 2 ];
SpiceDouble             root2 [ 2 ];

/.
Let's do one with imaginary roots just for fun.
./

rquad_c ( 1.,  2.,  3.,  root1,  root2 );

printf ( "Root #1: %12.7f %12.7f\n", root1[0], root1[1] );
printf ( "Root #2: %12.7f %12.7f\n", root2[0], root2[1] );

return ( 0 );
}

When this program was executed on a Mac/Intel/cc/64-bit
platform, the output was:

Root #1:   -1.0000000    1.4142136
Root #2:   -1.0000000   -1.4142136
```

Restrictions

```   1)  No checks for overflow of the roots are performed.
```

Literature_References

```   None.
```

Author_and_Institution

```   N.J. Bachman        (JPL)
J. Diaz del Rio     (ODC Space)
```

Version

```   -CSPICE Version 1.0.1, 04-AUG-2021 (JDR)

Edited the header to comply with NAIF standard.

Created complete code examples from existing code fragments.

-CSPICE Version 1.0.0, 13-JUN-1999 (NJB)
```

Index_Entries

```   roots of a quadratic equation
```
`Fri Dec 31 18:41:11 2021`