 # Inequalities – Square inequality – Exercise 5692

Exercise

Solve the square inequality:

$$-18x^2+30x\geq 13$$

There is no real solution

Solution

$$-18x^2+30x\geq 13$$

Move everything to one side:

$$-18x^2+30x-13 \geq 0$$

It is a square inequality. its coefficients are

$$a=-18, b=30, c=-13$$

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 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

$$x_{1,2}=\frac{-30\pm \sqrt{30^2-4\cdot (-18)\cdot (-13)}}{2\cdot (-18)}=$$

$$=\frac{-30\pm \sqrt{-36}}{-36}=$$

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 below the x-axis. Hence, there is no real solution to the inequality.

The graph of the equation:

$$y=-18x^2+30x-13$$

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

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