identify the intervals on each quadratic function is positive

Question

also how do you put it in vertex form to find the intervals equation please and thankyou.

y=x^2+9x+18

y=x^2+2x-8

y=x^2-5x-24

y=-x^2+4x+12

y=2x^2+12x+18

y=5x^2-3x-8

Answer 1

y = x² + 9x + 18 = (x +4.5)² - 2.25 > 0 for x < - 6 or x > -3

y = x² + 2x - 8 = (x + 1)² - 9 > 0 for x <- 4 or x > 2

y = x² - 5x - 24 = (x - 2.5)² - 30.25 > 0 for x < - 3 or x > 8

y = x² + 4x + 12 = (x + 2)² + 8 > 0 for - ∞ < x < ∞

y = 2x² + 12x + 18 = 2(x + 3)² > 0 for all real x except - 3

y = 5x² - 3x - 8 = 5(x - 0.3)² - 8.45> 0 for x < - 1 or x > 1.6

Answer 2

The graph of y = Ax²+Bx+C is a parabola opening upward if A > 0. If A<0 the parabola opens downward.

So, the graph of y = x²+9x+18 = (x+3)(x+6)=0 when x = -3 or -6. These are the x-intercepts. The graph opens upward. The function is positive when x < -6 and when x > -3.

You can do the second, third, fourth, and last problem in a similar manner.

For y = 2x²+12x+18, we have y = 2(x²+6x+9) = 2(x+3)², which is positive when x < -3 and when x > -3.