# How do you solve y=-x^2+2x-3 and y=x-5?

Sep 6, 2015

(-1, - 6)
(2, - 3)

#### Explanation:

$y = - {x}^{2} + 2 x - 3$ ------------------(1)
This is a downward facing U shaped curve
y = x - 5 -------------------------(2)
Straight line cuts it at two points.

Substitute $y = - {x}^{2} + 2 x - 3$ in equation (2)

$- {x}^{2} + 2 x - 3 = x - 5$
$- {x}^{2} + 2 x - 3 - x + 5 = 0$
$- {x}^{2} + x + 2 = 0$
$- {x}^{2} + 2 x - x + 2 = 0$
x ( - x + 2) + 1 (- x +2) = 0
(x + 1) ( - x + 2) = 0
(x + 1 = 0
x = -1

• x + 2 = 0
x = 2
Substitute the x values in equation (1)
At x = - 1 ; y = -1 - 5 = - 6

(- 1, -6) one point

At x = 2 ; y = 2 - 5 = 3

(2, -3) is another point