Inequalities – Square inequality – Exercise 5710

Exercise

Solve the square inequality:

$$x^2+3x+5>0$$

All x

Solution

$$x^2+3x+5>0$$

It is a square inequality. its coefficients are:

$$a=1, b=3, c=5$$

The coefficient of the squared expression (a) is positive, so the parabola (quadratic equation graph) “smiles” (= bowl-shaped). The sign of the inequality means we are looking for the sections the parabola is above the x-axis. We find the solutions (= zeros = roots) of the quadratic equation using the quadratic formula. Putting the coefficients in the formula gives us

$$x_{1,2}=\frac{-3\pm \sqrt{3^2-4\cdot 1\cdot 5}}{2\cdot 1}=$$

$$=\frac{-3\pm \sqrt{-11}}{2}$$

We got a negative number inside the root, so there is no real solution to the quadratic equation, i.e. its graph does not pass through the x-axis. Because she is “smiling”, she is always above the x-axis. Hence, the solution of the inequality is all x.

The graph of the equation:

$$y=x^2+3x+5$$

looks like this:

You can see that the graph is indeed above the x-axis for each x.

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