How to find the roots of a quadratic equation using the DM15L

 How to find the roots of a quadratic equation using the DM15L

Polynomial expression are best solved by applying Horner's method (Page 79 in HP15C's Owners Manual). Since the DM15L is so incredibly fast, speed is not such an issue compared with the original HP15C. Repeated calls to the same copy of the <em>x</em> storage register is prevented, thus speeding up the calculation. You might not notice the different in quadratic polynomials but there is a difference in the cubic and fourth degree polynomials. The sample below is a simple quadratic expression and easy to solve with the DM15L.

$$  3-x=2x^2 $$

That format is no good for the solver and must be set equal to zero.

$$ 2x^2-3+x=0 $$ 

Still not right, the exponents must be removed for optimal speed of the processor. Not that it really matters on the DM15L, but it is much more efficient.

$$  x(2x+1)-3=0$$ 

The short keystroke program follows:

[f][R↓]
[g][R/S]
[f][SST][0]
[2]
[×]
[1][+]
[×]
[3][-]
[g][GSB]
[g][R/S]

To find the two roots, we have to enter two sets of estimates that we think the answer lies in. I will use -10 to 0 and 0.1 to 10. Quite wide but the DM15L is so fast the answers are nearly instantaneous. 

-10 [ENTER]
0[F][SOLVE][0]

The answer should be -1.5

To get the other root we enter the following:

0.1 [ENTER]
10[F][SOLVE][0]

The answer should be 1.0.

Is this answer right. Lets test them with the formula for positive roots; 

$$  x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} $$

$$  a=2, b=1, c=-3$$

If we substitute the values in the equations below:

$$  x = \frac{-b + \sqrt{b^2-4ac}}{2a} $$

$$ x = \frac{-1 + \sqrt{1^2-4 \times 2 \times -3}}{2 \times 2}  $$

$$  x = \frac{-1 + \sqrt{25}}{4}  $$

$$  x = \frac{-1 + 5}{4} $$

$$ x = \frac{4}{4}  $$

$$  x=1$$ 

Is this answer right. Lets test them with the formula for negative roots;  

$$ x = \frac{-b - \sqrt{b^2-4ac}}{2a} $$

$$ x = \frac{-1 - \sqrt{1^2-4 \times 2 \times -3}}{2 \times 2}  $$

$$ x = \frac{-1 - \sqrt{25}}{4}  $$

$$ x = \frac{-1 - 5}{4}  $$

$$ x = \frac{-6}{4}  $$

$$ x=-1.5 $$