Mathematico | Mathematico

Find the sum of the $x \in Z$ for which points on the parabola

y = x^{2} - 4 x + 7

lie below the line

y = x + 6

Hint

You need to solve the inequality

x^{2} - 4 x + 7 < x + 6

Draw a sketch.

Solution

We need to solve the inequality

\begin{aligned} x^{2} - 4 x + 7 & < x + 6 \\ x^{2} - 5 x + 1 & < 0 \end{aligned}

The roots are

\begin{aligned} x & = \frac{5 \pm \sqrt{21}}{2} \\ x & \in {0.209 \dots, 4.79 \dots} \end{aligned}

We should sketch the parabola $y = x^{2} - 5 x + 1$ to find the solutions

quadratic inequality

We see that the solution to the inequality is

0.209 \dots < x < 4.79 \dots

and the integer solutions are $x = 1, 2, 3, 4$ .