The formula to calculate the First Root of a Quadratic Equation is:
\[ x_1 = \frac{-(b) + \sqrt{b^2 - 4ac}}{2a} \]
Where:
The First Root of a Quadratic Equation is the value of one of the variables satisfying the given quadratic equation \(f(x)\), such that \(f(x_1) = 0\).
Let's assume the following values:
Using the formula:
\[ x_1 = \frac{-(8) + \sqrt{8^2 - 4 \cdot 2 \cdot (-42)}}{2 \cdot 2} \]
Evaluating:
\[ x_1 = 3 \]
The First Root of the Quadratic Equation is 3.
| Numerical Coefficient a | Numerical Coefficient b | Numerical Coefficient c | First Root of Quadratic Equation |
|---|---|---|---|
| 1 | 5 | -10 | 1.531128874149 |
| 1 | 5 | -5 | 0.854101966250 |
| 1 | 5 | 0 | 0.000000000000 |
| 1 | 5 | 5 | -1.381966011250 |
| 1 | 5 | 10 | No real root |
| 1 | 6 | -10 | 1.358898943541 |
| 1 | 6 | -5 | 0.741657386774 |
| 1 | 6 | 0 | 0.000000000000 |
| 1 | 6 | 5 | -1.000000000000 |
| 1 | 6 | 10 | No real root |
| 1 | 7 | -10 | 1.216990566028 |
| 1 | 7 | -5 | 0.653311931459 |
| 1 | 7 | 0 | 0.000000000000 |
| 1 | 7 | 5 | -0.807417596433 |
| 1 | 7 | 10 | -2.000000000000 |
| 1 | 8 | -10 | 1.099019513593 |
| 1 | 8 | -5 | 0.582575694956 |
| 1 | 8 | 0 | 0.000000000000 |
| 1 | 8 | 5 | -0.683375209645 |
| 1 | 8 | 10 | -1.550510257217 |
| 1 | 9 | -10 | 1.000000000000 |
| 1 | 9 | -5 | 0.524937810560 |
| 1 | 9 | 0 | 0.000000000000 |
| 1 | 9 | 5 | -0.594875162047 |
| 1 | 9 | 10 | -1.298437881284 |
| 1 | 10 | -10 | 0.916079783100 |
| 1 | 10 | -5 | 0.477225575052 |
| 1 | 10 | 0 | 0.000000000000 |
| 1 | 10 | 5 | -0.527864045000 |
| 1 | 10 | 10 | -1.127016653793 |
| 2 | 5 | -10 | 1.311737691490 |
| 2 | 5 | -5 | 0.765564437075 |
| 2 | 5 | 0 | 0.000000000000 |
| 2 | 5 | 5 | No real root |
| 2 | 5 | 10 | No real root |
| 2 | 6 | -10 | 1.192582403567 |
| 2 | 6 | -5 | 0.679449471770 |
| 2 | 6 | 0 | 0.000000000000 |
| 2 | 6 | 5 | No real root |
| 2 | 6 | 10 | No real root |
| 2 | 7 | -10 | 1.089454172900 |
| 2 | 7 | -5 | 0.608495283014 |
| 2 | 7 | 0 | 0.000000000000 |
| 2 | 7 | 5 | -1.000000000000 |
| 2 | 7 | 10 | No real root |
| 2 | 8 | -10 | 1.000000000000 |
| 2 | 8 | -5 | 0.549509756796 |
| 2 | 8 | 0 | 0.000000000000 |
| 2 | 8 | 5 | -0.775255128608 |
| 2 | 8 | 10 | No real root |
| 2 | 9 | -10 | 0.922144385112 |
| 2 | 9 | -5 | 0.500000000000 |
| 2 | 9 | 0 | 0.000000000000 |
| 2 | 9 | 5 | -0.649218940642 |
| 2 | 9 | 10 | -2.000000000000 |
| 2 | 10 | -10 | 0.854101966250 |
| 2 | 10 | -5 | 0.458039891550 |
| 2 | 10 | 0 | 0.000000000000 |
| 2 | 10 | 5 | -0.563508326896 |
| 2 | 10 | 10 | -1.381966011250 |
| 3 | 5 | -10 | 1.173599096465 |
| 3 | 5 | -5 | 0.703257409549 |
| 3 | 5 | 0 | 0.000000000000 |
| 3 | 5 | 5 | No real root |
| 3 | 5 | 10 | No real root |
| 3 | 6 | -10 | 1.081665999466 |
| 3 | 6 | -5 | 0.632993161855 |
| 3 | 6 | 0 | 0.000000000000 |
| 3 | 6 | 5 | No real root |
| 3 | 6 | 10 | No real root |
| 3 | 7 | -10 | 1.000000000000 |
| 3 | 7 | -5 | 0.573384418152 |
| 3 | 7 | 0 | 0.000000000000 |
| 3 | 7 | 5 | No real root |
| 3 | 7 | 10 | No real root |
| 3 | 8 | -10 | 0.927443327708 |
| 3 | 8 | -5 | 0.522588120943 |
| 3 | 8 | 0 | 0.000000000000 |
| 3 | 8 | 5 | -1.000000000000 |
| 3 | 8 | 10 | No real root |
| 3 | 9 | -10 | 0.862907813126 |
| 3 | 9 | -5 | 0.479057014506 |
| 3 | 9 | 0 | 0.000000000000 |
| 3 | 9 | 5 | -0.736237384174 |
| 3 | 9 | 10 | No real root |
| 3 | 10 | -10 | 0.805399495699 |
| 3 | 10 | -5 | 0.441518440112 |
| 3 | 10 | 0 | 0.000000000000 |
| 3 | 10 | 5 | -0.612574113277 |
| 3 | 10 | 10 | No real root |