Solving Equations Graphically and Analytically
Analytical Methods
Analytical methods involve solving equations using algebraic techniques. These methods are particularly useful for equations that can be manipulated into standard forms.
Consider the equation $e^{2x} - 5e^x + 4 = 0$. This can be solved analytically by substituting $y = e^x$:
- Rewrite the equation: $y^2 - 5y + 4 = 0$
- This is a quadratic equation in $y$, which can be solved using the quadratic formula: $y = \frac{5 \pm \sqrt{25 - 16}}{2} = \frac{5 \pm 3}{2}$
- Therefore, $y = 4$ or $y = 1$
- Substituting back $e^x = 4$ or $e^x = 1$
- Taking natural logarithms: $x = \ln 4$ or $x = 0$
Thus, the solutions are $x = \ln 4$ and $x = 0$.
When solving exponential equations analytically, try to isolate the exponential term and then use logarithms to solve for the variable.
Graphical Methods
Graphical methods involve plotting the functions on both sides of the equation and finding their intersection points. This approach is particularly useful when analytical methods are difficult or impossible to apply.
To solve $e^x = \sin x$ graphically:
- Plot $y = e^x$ and $y = \sin x$ on the same coordinate system
- Identify the points where the graphs intersect
The intersection points give the solutions to the equation.
Graphical methods often provide approximate solutions. For more precise values, use technology. Alternatively, be smart and use analytic techniques.
Using Technology
Modern graphing calculators and software can solve complex equations quickly and accurately. This is especially useful for equations that are difficult to solve analytically or graphically by hand.
To solve $x^4 + 5x - 6 = 0$ using technology:
- Input the equation into a graphing calculator or software
- Use the equation solver or root-finding function
- The technology will display the solutions, which are approximately: $x \approx -1.5605, -0.3765, 1.1937$
Don't rely solely on technology without understanding the underlying principles. It's important to interpret results and check if they make sense in the context of the problem.
If you get a mass of $5$ billion kilograms, I think you should recheck your working.
