Question

Using graphing, what is the approximate solution of this equation?

Answer 1

Answer:

  D.  x ≈ 2.60

Step-by-step explanation:

The equation we have graphed is the one shown in the attachment. It is slightly different from the one in this problem statement.

Graphical solution

It is often convenient to find a graphical solution to an equation by writing it in the form ...

  f(x) = 0

Then the solutions are the x-intercepts of the graph, points that most graphing calculators can readily identify.

We have done this here by defining ...

  $y_1=3x^2-6x-4\\\\y_2=-\dfrac{2}{x+3}+1$

The graph is of y₁ -y₂. X-values where y₁-y₂ = 0 are solutions to the original equation, y₁ = y₂.

This equation is seen to have three solutions, approximately ...

  x = {-3.05, -0.55, 2.60}

Of these, only x ≈ 2.60 is listed among the answer choices.

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Additional comment

The value of y₁ -y₂ expressed in standard form is ...

  $3x^2-6x-4+\dfrac{2}{x+3}-1=0\\\\\dfrac{(x+3)(3x^2-6x-5)+2}{x+3}=0\\\\\dfrac{3x^3+3x^2-23x-13}{x+3}=0\\\\3x^3+3x^2-23x-13=0\quad(x\ne -3)$

The irrational solutions to this cubic can be found "exactly" by any of several applicable formulas. Numerical solutions are almost trivial to find with appropriate technology.

  x ≈ {−3.04857564747, −0.547524627423, 2.5961002749}.