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Chapter 2: Problem 36

Explain what is unusual about the solution set of the inequality. $$x^{2}+3 x+8>0$$

Short Answer

Expert verified
What's unusual is that the inequality \(x^{2}+3x+8 >0\) has no real roots, making it true for all real x. It's because the discriminant of the quadratic inequality is negative, indicating that the roots are complex. This is unusual as typically quadratic inequalities have real solutions.

Step by step solution

01

Find the Discriminant

The discriminant of a quadratic expression \(ax^{2} + bx + c\), is given by the formula \(D=b^{2}-4ac\). Therefore, for the given quadratic \(x^{2}+3x+8\), calculate the discriminant. Substitute \(a=1\), \(b=3\), and \(c=8\) into the discriminant formula, and calculate the value of D. If D is greater than 0, it means there are two distinct real roots. If D equals 0, there is exactly one real root (roots are identical). If D is less than 0, there are two distinct complex roots (not real).
02

Calculate the Roots

If the discriminant is greater than or equal to zero, then calculate the roots of the quadratic inequality using the quadratic formula \(x_{1,2} = \frac{-b \pm \sqrt{D}}{2a}\). Plug in the values of a, b, and D to get the roots. You already know the type of roots from step 1: if there are two real roots, there should be two x-values from this formula, if there is one real root, there should be one x-value, and if there are no real roots, then the solution to this formula will provide a complex number.
03

Analyze the Solution Set

Finally, determine whether the solutions are unusual. If the solutions are complex, this is considered 'unusual,' because quadratic inequalities don't typically have complex solution sets when we are considering real numbers. Alternatively, you can also plot the graph of \(y=x^2+3x+8\), and observe the intervals where it is greater than 0. The solution set of the inequality should be these intervals. If it's not possible to find such intervals, again, that's unusual and means the inequality is true for all real x.

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