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Algebra and Precalculus A Practical Guide

Cameron Geisler

Section 17.3 Solving Equations Numerically with a Calculator

A large amount of high school mathematics is about solving equations. You learn various techniques to solve equations by hand (also called solving algebraically), which work for various types of equations. However, for many types of equations, there is no algebraic technique that can solve them.
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Example 17.3.1.

For example, in fact, these equations can’t be solved algebraically:
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  • \(\displaystyle x^5 - x - 1 = 0\)
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  • \(\displaystyle x^7 - 2x^3 + 4 = 0\)
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No matter how hard you try to manipulate the equation, factor, and isolate for \(x\text{,}\) you won’t be able to solve for \(x\) exactly.
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If you have learned about exponential functions, logarithmic functions, and/or trigonometric functions, here are some more examples:
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  • \(\displaystyle 2^x = x\)
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  • \(\displaystyle e^x + x = 0\)
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  • \(\displaystyle e^x = x^2\)
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  • \(\displaystyle \log{x} = \frac{1}{x}\)
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  • \(\displaystyle \cos{x} = x\)
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  • \(\displaystyle \sin{x} = x^2\)
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These equations are simple to write down, but none of them can be solved algebraically. More generally, most equations with a combination of exponentials, polynomials, and trigonometric functions are unsolvable algebraically.
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In this case, the next best thing is to find an approximate numerical solution.
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These make up the two broad approaches to solving equations:
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  • Algebraic / analytic / exact methods: Manipulating the equation using algebraic rules, to isolate the variable, and find an exact solution.
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  • Approximate / numerical methods: Finding a solution which is an approximate decimal number, that is close enough.
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Numerical solutions are useful in a few situations:
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  • If an exact solution can’t be found (it’s just impossible).
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  • If an exact solution is possible but just really difficult to find (especially in applications, an approximate numerical answer is usually good enough)
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  • If you’ve already solved algebraically, and you want to check that your final answer is correct.
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Subsection 17.3.1 Solving Using a Scientific Calculator

My recommended calculator Casio fx-991ES Plus C 2nd edition (or some other similar scientific calculators) can solve equations numerically. This is truly powerful, because you can check your solutions to any equation.
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  1. First, make sure you’re in COMP (computation) mode (which is the regular mode), by pressing MODE, then select COMP.
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  2. Enter your equation. Use the ALPHA button to get X (look for the button that has “X” above it). For the \(=\) sign, use ALPHA then CALC.
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  3. Start the SOLVE function, by pressing SHIFT, then CALC.
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  4. Provide an initial guess for \(x\text{,}\) such as \(1\text{,}\) then press \(=\text{.}\)
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  5. View the solution. The calculator shows the value of \(x\text{.}\) It also displays \(L-R\) (left side minus right side), which will be 0 (or at least very close to \(0\)) if the solution is accurate.
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Example 17.3.2. Solving a Radical Equation.

For example, to solve \(\sqrt{2x+5}=4x\text{,}\)
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  1. Enter \(\sqrt{2X+5}=4X\text{.}\)
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  2. Select SOLVE, by pressing SHIFT, then CALC.
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  3. Enter an initial guess, like \(1\text{,}\) and press \(=\text{.}\)
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The solution should be \(x=0.625\text{.}\)
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  • If the solution is not a “nice” number (like a whole number), the solution will be a decimal like \(1.414\text{,}\) not symbolic like \(\sqrt{2}\text{.}\)
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  • If \(L-R\) is anything but 0 (or very close to it), then the calculator couldn’t find the solution. Try a different guess.
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The initial guess kinda matters, but most reasonable guesses should give you the correct answer. Here are some broad principles:
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  • Ideally, try to give a good guess as you can. For example, a number close to the solution you got from solving by hand.
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  • If there are multiple solutions, the calculator will only give you one of them (each time). This is because the calculator basically “pulls” your initial guess towards the closest solution. E.g. if the solutions are \(-2\) and \(8\text{,}\) and your initial guess is \(7\text{,}\) you will most likely get \(x=8\text{.}\) Whereas if you choose initial guess of \(0\text{,}\) you will get \(x=-2\text{.}\)
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  • If you have no idea what the solution should be, try 0 or 1 as a default. Because many equations have solutions near 0.
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This feature is truly powerful, because you can check your solutions to any equation.
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  • If the calculator agrees with your algebraic steps, then you know your answer is correct.
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  • If it gives a different solution than what you found, then you know you have a mistake somewhere, and can go back and try to find it. E.g. maybe you dropped a negative sign, or didn’t collect like terms properly.
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You can also use it to solve an equation if you are stuck, however most tests will require you to show your work as well.
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