First, expand the term squared on the right side of the equation using the rule:
(color(red)(a) + color(blue)(b))^2 = color(red)(a)^2 + 2color(red)(a)color(blue)(b) + color(blue)(b)^2
Substituting x for a and 2 for b gives:
y = -2(color(red)(x) + color(blue)(2))^2 + 16x^2 - x - 5
y = -2(color(red)(x)^2 + (2 * color(red)(x) * color(blue)(2)) + color(blue)(2)^2) + 16x^2 - x - 5
y = -2(x^2 + 4x + 4) + 16x^2 - x - 5
Now, expand the term in parenthesis, group and combine like items:
y = (-2 * x^2) + (2 * 4x) + (2 * 4) + 16x^2 - x - 5
y = -2x^2 + 8x + 8 + 16x^2 - x - 5
y = -2x^2 + 16x^2 + 8x - x + 8 - 5
y = (-2 + 16)x^2 + (8 - 1)x + (8 - 5)
y = 14x^2 + 7x + 3
We can now use the quadratic equation to solve this problem. The quadratic formula states:
For ax^2 + bx + c = 0, the values of x which are the solutions to the equation are given by:
x = (-color(blue)(b) +- sqrt(color(blue)(b)^2 - (4color(red)(a)color(green)(c))))/(2 * color(red)(a))
Substituting:
color(red)(14) for color(red)(a)
color(blue)(7) for color(blue)(b)
color(green)(3) for color(green)(c) gives:
x = (-color(blue)(7) +- sqrt(color(blue)(7)^2 - (4 * color(red)(14) * color(green)(3))))/(2 * color(red)(14))
x = (-color(blue)(7) +- sqrt(49 - 168))/28
x = (-color(blue)(7) +- sqrt(-119))/28
Or
x = -color(blue)(7)/28 +- sqrt(-119)/28
x = -1/4 +- sqrt(-119)/28
