Can you help me with this problem?

Question

I need to graph this quadratic inequality. Can you walk me through it?

y is negative or equal to -2x^2-2x+1

Answer 1

by "y is negative or equal" do you mean y ≤ ? If so,

solve -2x² - 2x + 1 = 0 to get the zeros of the function:

-2x² - 2x = -1
2x² + 2x = 1
2(x² + x + 1/4) = 1 + 2(1/4)
2(x + 1/2)² = 3/2
(x + 1/2)² = 3/4
x + 1/2 = ±(√3)/2
x = -1/2 ± (√3)/2

Since the graph is a parabola that opens down, y ≤ 0 when x ≤ left zero and y ≤ 0 when x ≥ right zero, so your solution set is

x ≤ -1/2 - (√3)/2 OR x ≥ -1/2 + (√3)/2.

Answer 2

I think you mean "less than or equal to...".

-2x^2 - 2x + 1 is an inverted parabola with a vertex maximum at (-b/2a) x = -1/2. b^2 - 4ac = 4 + 8 = 12, giving roots (where it will cross the x-axis) of sqrt(3) - 1/2 and -sqrt(3) - 1/2. Everything below (inside) and including the curve is the solution.