Question

at which points are the equations $y = x^2 + 5x - 2$ and $y = x + 1$ equal?

Explanation:

Step1: Set equations equal

$x^2 +5x -2 = x +1$

Step2: Rearrange to standard quadratic

$x^2 +4x -3 =0$

Step3: Solve quadratic via quadratic formula

$$x = \frac{-4\pm\sqrt{16 +12}}{2} = \frac{-4\pm\sqrt{28}}{2} = -2\pm\sqrt{7}$$

Step4: Find corresponding y-values

For $x=-2+\sqrt{7}$, $y = (-2+\sqrt{7}) +1 = -1+\sqrt{7}$
For $x=-2-\sqrt{7}$, $y = (-2-\sqrt{7}) +1 = -1-\sqrt{7}$

Answer:

The points are $\boldsymbol{(-2+\sqrt{7}, -1+\sqrt{7})}$ and $\boldsymbol{(-2-\sqrt{7}, -1-\sqrt{7})}$, or approximately $(0.646, 1.646)$ and $(-4.646, -3.646)$