Solve X³+6x²-40x=192 By Graphing

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Solve x³+6x²-40x=192 by Graphing

In this article, we're going to solve the equation x³ + 6x² - 40x = 192 by using a graphical approach. This involves graphing two separate equations, y = x³ + 6x² - 40x and y = 192, and finding their intersection points. The x-coordinates of these intersection points will give us the solutions to the original equation. So, grab your graphing tools or software, and let's dive in!

Understanding the Problem

Before we jump into graphing, let's make sure we understand what we're trying to achieve. We have a cubic equation, which means it can have up to three real solutions. Graphing is a visual way to find these solutions. We split the original equation into two separate equations:

  1. y = x³ + 6x² - 40x (a cubic function)
  2. y = 192 (a horizontal line)

The solutions to the original equation are the x-values where these two graphs intersect. Essentially, we're looking for the points where the cubic function's y-value is equal to 192.

Why this approach? Well, solving cubic equations algebraically can be a bit of a headache. Graphing provides a more intuitive and visual method, especially when you have access to graphing tools or software. It allows us to quickly identify the real roots of the equation.

Graphing the Equations

Okay, guys, let's get to the fun part: graphing! You can use a graphing calculator, online graphing tools like Desmos or Geogebra, or even good old-fashioned graph paper if you're feeling old-school.

Graphing y = x³ + 6x² - 40x

This is a cubic function, so its graph will have a characteristic S-shape. To get a good graph, plot several points. Let's analyze this function to understand its behavior:

  • End behavior: As x approaches positive infinity, y also approaches positive infinity. As x approaches negative infinity, y approaches negative infinity.
  • Roots: We can factor out an x from the equation: y = x(x² + 6x - 40). Further factoring gives us y = x(x + 10)(x - 4). This tells us the graph crosses the x-axis at x = -10, x = 0, and x = 4. These are the x-intercepts.
  • Turning Points: Cubic functions have up to two turning points (local maxima and minima). To find these precisely, you'd typically use calculus (derivatives), but for our graphical approach, we can estimate them by plotting points and observing where the graph changes direction.

Now, let's plot some points. Choose x-values around the roots we found (-10, 0, and 4) to get a good picture of the curve. For example:

  • x = -12: y = (-12)³ + 6(-12)² - 40(-12) = -1728 + 864 + 480 = -384
  • x = -10: y = 0 (x-intercept)
  • x = -8: y = (-8)³ + 6(-8)² - 40(-8) = -512 + 384 + 320 = 192
  • x = -6: y = (-6)³ + 6(-6)² - 40(-6) = -216 + 216 + 240 = 240
  • x = -4: y = (-4)³ + 6(-4)² - 40(-4) = -64 + 96 + 160 = 192
  • x = 0: y = 0 (x-intercept)
  • x = 2: y = (2)³ + 6(2)² - 40(2) = 8 + 24 - 80 = -48
  • x = 4: y = 0 (x-intercept)
  • x = 6: y = (6)³ + 6(6)² - 40(6) = 216 + 216 - 240 = 192

Plot these points and connect them with a smooth curve to get the graph of y = x³ + 6x² - 40x. It should look like an S-shaped curve that crosses the x-axis at -10, 0, and 4.

Graphing y = 192

This is super easy! y = 192 is simply a horizontal line that crosses the y-axis at 192. Draw this line on the same graph as the cubic function.

Finding the Intersections

Now, the moment of truth! Look at your graph and identify where the cubic function y = x³ + 6x² - 40x intersects the horizontal line y = 192. You should see three intersection points. We already calculated these points when generating values to plot the cubic function. Looking at the points we already calculated:

  • When x = -8, y = 192
  • When x = -4, y = 192
  • When x = 6, y = 192

Therefore, the cubic function intersects the line y = 192 at x = -8, x = -4, and x = 6.

Identifying the Solutions

The x-coordinates of the intersection points are the solutions to the equation x³ + 6x² - 40x = 192. From our graph (and our calculations), we found three intersection points:

  • x = -8
  • x = -4
  • x = 6

So, the solutions to the equation are x = -8, x = -4, and x = 6.

Verifying the Solutions

To be absolutely sure, let's plug these values back into the original equation:

  • For x = -8:
    • (-8)³ + 6(-8)² - 40(-8) = -512 + 384 + 320 = 192 (Correct!)
  • For x = -4:
    • (-4)³ + 6(-4)² - 40(-4) = -64 + 96 + 160 = 192 (Correct!)
  • For x = 6:
    • (6)³ + 6(6)² - 40(6) = 216 + 216 - 240 = 192 (Correct!)

All three values satisfy the original equation. Fantastic!

Conclusion

By graphing the system of equations y = x³ + 6x² - 40x and y = 192, we successfully found the solutions to the equation x³ + 6x² - 40x = 192. The solutions are x = -8, x = -4, and x = 6. Graphing is a powerful tool for visualizing and solving equations, especially when algebraic methods are complex. This method provides a clear and intuitive way to find the real roots of the equation. So next time you are presented with a similar problem, consider using graphs! I hope this helped, good luck guys!