Question

3x - x^{2} + 2 < 0
solve this quadratic inequality graphically

Answer 1

(−∞,−3) (2,+∞)

Step-by-step explanation:

graph the parabola

y=−x2−x+6

and consider which parts are less than zero, that is below the x-axis finding the x and y intercepts

let x = 0, in the equation for y-intercept

let y = 0, in the equation for x-intercepts

x=0→ y=6←

y-intercept

y=0→ −x2−x+6=0

multiply through by - 1

⇒ x2+x−6=0

the factors which multiply to give - 6 and sum to + 1 are + 3 and - 2

⇒(x+3)(x−2)=0

⇒x=−3 or x=2←

x-intercepts

obtaining the shape of the parabola

if a>0

then minimum if a < 0

then maximum for y =−x2−x+6xa<0

we can now graph the parabola

graph{-x^2-x+6 [-10, 10, -5, 5]}

x<−3 or x>2

are the parts below the x-axis

in interval notation

(−∞,−3)∪(2,+∞)