# Inequalities – Square inequality – Exercise 5707

Exercise

Solve the square inequality:

$$6x-x^2<10$$

All x

Solution

$$6x-x^2<10$$

Move everything to one side:

$$6x-x^2-10<0$$

It is a square inequality. its coefficients are:

$$a=-1, b=6, c=-10$$

The coefficient of the squared expression (a) is negative, so the parabola (quadratic equation graph) “cries” (= inverted-bowl-shaped). The sign of the inequality means we are looking for the sections the parabola is below 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

$$x_{1,2}=\frac{-6\pm \sqrt{6^2-4\cdot (-1)\cdot (-10)}}{2\cdot (-1)}=$$

$$=\frac{-6\pm \sqrt{-6}}{-2}$$

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 it is “crying”, it is always under the x-axis. Hence, inequality exists for every x.

The graph of the equation:

$$y=-x^2+6x-10$$

looks like this: You can see that the graph is indeed below the x-axis for all x.

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