Identify the​ interval(s) on which the quadratic function is positive

Question

y=5x^2-43x+56

Answer 1

So to solve this it helps to graph it. Since the number in front of the x² is positive, we know this is an upward facing parabola (an upward U shape). Part of it will be above and part will be below the x-axis. To find these ranges, we must solve for y = 0

first factor the equation:

y = (5x- 8)(x - 7)

solve for x: 5x - 8 = 0, x - 7 = 0

x = 8/5, x = 7

since it is an upward shaped U, the equation is positive when x is less than 8/5 and when x is greater than 7.

this can be checked by simple substitution:

less than 8/5:

lets say x = 1

y = 5*(1)² - 43*1 + 56 = 18, a positive number

greater than 7:

let’s say x = 10

y= 5(10)²-43*10+ 56= 14, a positive number

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Answer 2

Hi Alyssa, one way to solve this problem would be to set the equation equal to zero and solve for x using the quadratic equation. In doing so you could find that x = 1.6 and x=7. After finding these as your points, you could use a graphing utility like Desmos or a graphing calculator to determine whether the graph of the function is positive or negative near those points. After doing so, you can determine that the function is positive on the intervals (-∞,1.6) and (7,∞). I hope this helps! Let me know if there is any confusion!

Answer 3

(5x-8)(x-7) >0

5x-8=0,

x-7=0

x =7, 8/5 are the boundaries of the intervals

(-infinity, 8/5)U(7,infinity) in interval notation

or

x<8/5 or x>7

{x|x<8/5, x>7}