[latex]2x^3 - 4x = 5x^2 - 3[/latex]
Hint.
Try small integer values for [latex]x{.}[/latex]
Solution
[latex]x = 3[/latex]
Chapter 5: Equations and Identities
Each of the following “solutions” contains an error. Find the error and supply a correct solution.
[latex]\underline{\qquad\qquad\qquad\qquad}[/latex]
It is important to distinguish between an algebraic expression and an equation. An equation is a statement that two algebraic expressions are equal. It may be true or false, depending on the values of any variables involved. Here are some examples of equations.
[latex]5(2 + 6) = 5(2) + 5(6)[/latex]
[latex]\sqrt{3^2} + \sqrt{4^2}= 3 + 4[/latex]
[latex]x^2 + 3x = 10[/latex]
The first equation is true, the second is false, and the third equation is true only if [latex]x = 2[/latex] or [latex]x = -5{.}[/latex] When you solve an equation, you are finding the values of the variable that make the equation true.
Checkpoint 5.20.
Try small integer values for [latex]x{.}[/latex]
[latex]x = 3[/latex]
You probably remember a number of algebraic techniques for solving equations of different types. Another useful equation-solving method uses graphs.
Use a graph to solve the equation [latex]x^3 - 2x^2 - 5x = -6{.}[/latex]
We graph the expressions on either side of the equation; that is, we graph [latex]y = x^3 - 2x^2 - 5x[/latex] and [latex]y = -6[/latex] on the same grid, as shown below.
We are looking for any values of [latex]x[/latex] where the two [latex]y[/latex]-values are equal, and these occur at the intersection points of the two graphs. At those points, the [latex]x[/latex]-values are [latex]x = -2,~ x = 1[/latex] and [latex]x = 3{,}[/latex] and these are the solutions of the equation. You can check that all three values make the equation true.
Checkpoint 5.22.
The graph does not cross the line [latex]y = 0{.}[/latex]
The first Ferris wheel was built for the Chicago World's Fair in 1893. It had a diameter of 250 feet and could carry 2160 people in 36 carriages. From the top of the wheel, passengers could see into four states. After loading all the passengers, the wheel made one revolution in nine minutes.
If you are in the bottom carriage of the Ferris wheel at the start of its revolution, your height after [latex]t[/latex] seconds is given by
[latex]h = f(t) = 139 - 125\cos \left(\dfrac{2t}{3}\right)[/latex]
For how long are you more than 240 feet above the ground?
The figure below shows a graph of the height function and a horizontal line at [latex]h = 240{.}[/latex]
From the graph, we see that [latex]h = 240[/latex] at approximately 215 seconds and 325 seconds into the ride. Your height is more than 240 feet between those two times, or for about 110 seconds.
In the example above, we used a graph to solve the equation [latex]h = 240{,}[/latex] or
[latex]139 - 125\cos \left(\dfrac{2t}{3}\right) = 240[/latex]
To find a more precise solution, we can use algebraic methods. As an example, we'll solve the slightly simpler equation
[latex]139 - 125\cos\t