Mathematico | Mathematico

The points $(3, p)$ and $(q, 1)$ lie on the parabola

y = x^{2} - 2 x - 9

Find the possible values of $p$ and $q .$

Give the sum of the squares of all solutions.

Hint

Substitute $x = 3$ to find $p .$

Substitute $y = 1$ to find $q .$

Solution

To find $p$ , we substitute $x = 3$

\begin{aligned} p & = 3^{2} - 2 \times 3 - 9 \\ = - 6 \end{aligned}

To find $q$ , we substitute $y = 1$

\begin{aligned} 1 & = x^{2} - 2 x - 9 \\ x^{2} - 2 x - 10 & = 0 \\ x & = \frac{2 \pm \sqrt{4 - 4 \times 1 \times (- 10)}}{2 \times 1} \\ = \frac{2 \pm \sqrt{44}}{2} \\ = \frac{2 \pm \sqrt{11}}{2} \\ = 1 \pm \sqrt{11} \end{aligned}